Reading up a bit on NMT, I question the "strong support" overall in New Mexico. As was the wont of the slaveholders and their supporters, they count only the interests and views of "white" people, in this context specifically English speaking ones, as the only thing that matters. As Taney said, people of color have no rights a "white man is bound to respect;" this is their motto....The reason Arizona is included is the strong support for the Confederates there, and the fact that it turns the confederate states into a sort of buffer between any potential Mexican revanchists and the Union.
But it seems to me if we count people as people, "Arizona" as the secessionists defined it (based to be sure on a traditional definition of Arizona as a region which overlapped the proposed southern splinter half of NMT and comprised about 2/3 or so of that more rectilinear section of NMT) had some inhabitants, and more importantly other peoples nearby had some vital interests in it, that would ill dispose the US Congress (which has Constitutional authority over organization of US Territories) to grant the petition of even an overwhelming local majority in key areas. Broadly speaking, the demographics of NMT amounted to Native peoples who were admittedly not friendly to either the USA or CSA; the Spanish New Mexicans, an intermarried bunch of descendants of both regional Native people and Spanish settlers centered on Santa Fe and comprising a significant population, and somewhat diverse recent Anglo settlers. Important categories of the latter, especially in the southern tier of NMT, were indeed southerners. Gadsden himself was a South Carolina railroad owner and his vision was to develop a southern transcontinental route. Looking at the topography of the modern state of Arizona it seems that for any future rail connection of the securely Union parts of NMT--initially the northeast, the highly Hispanicized Rio Grande upper course and Pueblo Indian zones; if Mexican intervention were a thing to be concerned with it might be otherwise but these zones have populations whose sympathy for the Secessionist cause would be measured in negative numbers, having had bad experiences with Texan ambitions--to the west coast would have to run through the southern tier of NMT to the west. Assuming as I do a "go in peace" scenario, the boundary of Texas is the hard western frontier of the CSA, unless one argues they would seize land from NMT (or farther north, like Colorado)--whether Indian Territory goes north or south has very little bearing here. However biased toward participating in CSA the Anglo settlers of the whole NMT are, and however concentrated in the southern "CSA Arizona" reach they might be, they are not alone--the Union as a whole has collective interests in retaining all of NMT, the Anglos from the North and midwest of the former complete Union are not secessionist, and the Spanish speaking and Native peoples of New Mexico have no secessionist sympathies. The latter have no Unionist ones either; even if my rosy hopes of a rapprochement based on more ethical standards being met by US authorities in the future toward Native peoples generally are realized, in the crucial months of presumably late 1860 and early 1861 we have to assume the Native people are on a spectrum of indifferent to hostile to both nations. But the immediate interest of Native people of New Mexico, whatever the possible attractions of secessionism and joining CSA might be among the Indian Territory residents to the east, is definitely opposed to the Texan interests that the purported lean of southern NMT is basically all about.
Looking at various maps it seems that in terms of non-Native people, the action in NMT is all concentrated to the east, with both Unionists and secessionists carrying out expeditions westward, but no substantial settlements of Anglos existing beyond lightly manned way stations to California as yet. It is not then inconceivable to me that a conciliatory US leadership might concede some swathes of southeastern NMT to the CSA, presumably conveying it specifically to Texan control, but they will most definitely not permit that concession to reach very far west.
One thing glancing at various Wikipedia pages on New Mexico in the Civil War, Confederate Arizona, Traditional Arizona, and the Gadsden Purchase bring up, is that OTL the withdrawal of US regular army troops eastward in early 1861 was a pivotal event. I cannot stress this enough--if the US government decides to adopt a no-war policy toward the secessionist states, thereby agreeing to recognize their legitimacy as newly independent states under the overarching sovereignty of the CSA, then this withdrawal eastward never takes place. At a minimum, the level of force the Federal government deemed necessary and reasonable to keep peace, order and security in the Southwest remains on station. I think that one response of the US government to secession will in fact be to raise levels of force, to call up volunteers, increase funding of regular army and navy, and reevaluate their prior estimates of force upward. Eventually then the New Mexico Territory contingents will be increased, but that might take a while--at the very least though they will not be withdrawn. It might be that the chain of command in both the Army and civil Territorial government is much infiltrated with individuals with strong Southern ties and presumably secessionist sympathies. But this cannot lead to coups purporting to hand over US territory to Texas or the CSA without others within the same structures, and various local interests, joining to reassert Union authority. The story of the Civil War fronts in the far West involved a new army moving in from Texas; if the CSA keeps the peace with the USA, that army cannot leave Texas. If it does, the CSA and Texan authorities will have to maintain plausible deniability and hold them to be lawless filibusters, and the Union can dispatch whatever level of force they deem prudent, with a much freer hand than OTL, to go in to reinforce loyal forces in New Mexico and on paper the CSA cannot object.
So it remains a question of forces on the ground when various states, most importantly for this frontier, Texas, secede. Despite the possibility of disloyalty of certain numbers of officers and men, I think the balance of armed authority in NMT, assuming no general withdrawal of regular Army eastward, is plenty to constrain the local sentiment among Anglos for joining the CSA to strictly legal channels. People here note that various legislatures, in California and in NMT, adopted measures to split their bailiwicks north and south, implying a very strong Confederate sympathy in the southern bits that would require major military action to override, and more importantly implying that the Unionists among them were indifferent to losing those southern tracts. But while I would like to learn more about the alleged amicable divorce of southern California from its northern reaches, I have already read and noted that Congress rejected the proposal for a southern Arizona on the grounds of the population of the proposed Territory (then of the USA, this being 1858 and '59) being too small to justify it. Considering the interests of the northeastern inhabitants of NMT as a whole, I think insofar as Congressional approval is required--and even with much larger numbers involved, it always is Constitutionally (consider the situation of Puerto Rico for instance) the answer from the Union capital wherever it might move to would be "no." My reading implies that the later (1863) formation of a different Arizona Territory, comprising the modern state plus tracts that went to Nevada ultimately, in an east west division, had something to do not only with a different concept of the salient regions to be administered but also a formal requirement to avoid appearing to retroactively endorse the rejected north-south division previously proposed, which the CSA OTL presumed to try to implement--not limited to that; the forces invading from Texas also sought to secure control into Colorado.
No one here is bound as I am binding myself to think only of no-war scenarios; if there is a brief civil war followed by Union coming to terms short of reconquest, that is a whole other sheaf of possibilities, and in it, with USA forces necessarily siphoned off east and a free hand for whatever force might accomplish, the prospects depend on the balance of those forces, with the Union handicapped in the region, at least in the short run, which is all a short Civil War gives them. But holding to the premise that both secessionists and unionists are led by people who prefer to avoid a full on civil war for expedient reasons, I think we can discount this secessionist interest in NMT as being trumped by the regular Army that remains, reinforced by local anti-secessionist interests especially when we don't restrict our consideration to just the intramural posturing of Anglo settlers.
The USA has zero positive reason to want to alienate any of NMT; at most it might find it prudent to throw the Texans a bit of concession to shut them up but I doubt even that can be allowed. Looking to the future, the then currently settled core of loyalist NMT is in the modern state of New Mexico's north, and if they are not to be a dead end, they need to retain access to the topographically desirable Gadsden Purchase zones for access to California by rail, whereas the secessionists, however numerous in the regions near El Paso, were nowhere on the ground out there, or no more so than others. The Union does not owe the CSA any concessions save for reasons of expedience and the CSA has no leverage without risking general war in the west.
The situation in southern California might be worthy exploring more; I know OTL there were pitched battles fought there in OTL. But again, regular US Army forces being withdrawn east might account for the ability of the secessionists to organize the army contesting Union authority OTL which again would not apply here.
The notion that the Union would want to throw any section of NMT or any other territory to the CSA to buffer off Mexico strikes me as downright absurd; clearly bordering CSA is much more problematic than bordering Mexico! The USA wants the former border as short as possible; the latter is much more easily managed.