Looking at a map right now:
In a scenario in which Mexico managed to occupy most of the Southern CSA, including Louisiana and the US was tangled up until the very end of the war (or was otherwise unable to initially assist.)...
I could see the US easily getting back VA, TN, SC and NC (probably AK too).
Mexico easily gets a hold of the Southwest territories: Otl's AZ and NM.
Now, if Mexico really stomped the CSA and the timeframe (though not the manpower and munitions) that the US donated was negligible, there may be a fight at the peace talks over the Deep South (including LA and FL).
I see two intriguing possibilities:
A. The CSA could be allowed to survive as a buffer state consisting of MS, AL, and GA (giving it a port at least). No one is happy, least of all the CSA - and future hostilities are damn near an inevitability.
B. If the US pushes hard and Mexico is generous, Mexico keeps all or most of FL (The panhandle?), and the southern halves of LA, MS, AL, and GA - giving the US the top halves over an arbitrary line - essentially cutting the South into halves. They both get ports, although Mexico takes the Gulf ports pretty handily.
Mexico takes the southern third of Texas, and the US takes the top square bit of it, incorporating it into the OK territory. Texas is allowed to remain an independent state (with a small number of port cities near LA) populated by CSA refugees and Texans, and has trouble stopping immigration by filibusterers from both the US and Mexico for years.
Hostilities are inevitable in this one as well. Sooner or later, the US is going to want its Gulf Ports back and both the US and Mexico will crave the failing Texas.
The next war would be pretty brutal, perhaps a civil war in Texas between Mexicans, USers, and the hybrid culture of longtime native Texans and their newcoming Confederates. This triggers a Second Mexican American War and all bets are off as to who comes out ahead (while the easy answer is the US, it's never all that easy. Mexico has had the increased port space and revenue to build quite the navy since the Mexican-Confederate War.)
If the US is the aggressor but cannot keep up and Mexico wins, there could very likely be a push for Mexico to claim most of the former CSA (maybe minus VA, TN, and the Carolinas.) The occupation of the former CSA by Mexico would be extremely difficult, but handled ideally could result in anyone not wanting to be a part of Mexico fleeing north and any Mexican farmer fed up with the arid conditions of original Mexico taking their place. After a generation or two, (say by the 1910's or so) Mexican control over what was once US then CS then US again and now Mexican would be if not ideal, than mostly manageable.
And going a little TOO far here, but...
After the Second Mexican-American War ends in a decisive and crippling Mexican victory, a far better off economically and militarily and slightly more Anglicized Mexico enters a golden age of World Power, while the US devastated by economic sanctions and a loss of territory, military strength, and manpower undergoes years of revolutions, with her former master the UK and her only friendly neighbor Canada providing the only aid. This forms a necessary series of economic alliances with (and burdens to) Canada, who will - along with the UK - by the mid twentieth century essentially own the US.
Meanwhile, Mexico flourishes, has all the best European allies, and its biggest concern other than intense political rivalries is the sheer magnitude of American immigrants stealing into Mexico by the hundreds or even thousands.
The former US may eventually bestow its intense independent streak and elements of its constitution to the US-Canadian culture that is being subtly and organically crafted, while sacrificing some fealty to the Crown as a dominion. A necessary evil for a once great nation now at the brink of collapse.
Eventually, of course, The United Provinces of Canada and America (or whatever) will become completely independent from the Crown of the UK and take on an identity dominated by Canadian culture but with plenty of aspects of Northern US. Canada and Mexico are the only two North American countries and they're both extremely successful.
Bingo: Mexico and Canada wanked. (Has that ever happened with a POD this late?!)