Again, the major problem is that this won't occur until after the majority of the American West is settled and the healing from the Civil War has progressed enough that US citizens are looking overseas. This gives you a relatively late window. The US has no desire to be acquiring easily lost lands overseas when they have vast expanses to the west. And, given the choice between annexing Ireland or Canada, the US would always take Canada. This gives you a rather narrow window. In order to minimize the damage done on the US by the Civil War, you need to have it be ended much more quickly. (note that the US did practically nothing major in foreign policy from 1865 to 1898. That is a period of 33 years, after all). If the war occurs earlier and the Union wins (or if it occurs a bit later and has less support from the Upper South/Texas), then the US could potentially be acting overseas by the 1870s. That would put it square in the timeline for the scramble for Africa. (Note that the US was invited to the Congress of Berlin, but did not attend).
Also take into account that the American Colonization Society's founding of Liberia was a private enterprise, not one sponsored by the government. That is historically what occurred with US overseas endeavors, for the most part. See the American claim in North Borneo/the Colony of Ellena/the Kingdom of Ambong and Maroodoo, or the US claim on the Washington/Marquesas Islands. None were ever sponsored.
Another problem is that, historically, Morocco was the first nation to recognize the US OTL; to this date, it is the only overseas country to have a US historic site within it. Morocco also, historically, looked fondly on the US and acknowledged the US's claims during the Barbary Wars, giving safe haven to US citizens and sailors. That will not change at all without a PoD prior to the revolution.
Sicily is something that did occur OTL; if I recall, there was a plebiscite that gained 40,000 signatures or such declaring that Sicily should secede from Italy and join the US. That is a tiny minority, and it would take either a longer war (and a longer period of time away from Italy) for Sicily to make such a play, or for another rather large PoD).
Albania joining is something that I've discussed in a thread before; those stories seem largely apocryphal. I found two separate newspapers talking about word on the street, but I never heard of a resolution calling for it. The attempts were rather recent, in the 1990s, and there definitely was talk of it. There simply wasn't any official action on record I could locate. This has slightly more basis than jokes/talk about Denmark or Poland joining the US because of relative fondness for the nations, which is hearsay at best.
Just to clarify, the US offered Denmark over a billion dollars in WW2 for Greenland; the Danish refused to sell. This will not go through unless Denmark is in far worse shape (longer occupation combined with US higher offers), or if WW2 goes into a stalemate as in AANW, and Greenland becomes de facto annexed.
Now, the situation would have to be engendered: you must have the US desire some territory on that rim in some fashion. How, though? (I mean, Jan Mayan could count as a Greenland territory, or perhaps Svalbard...) but, realistically, it must be part of some other interference in Europe within the time frame of 1870 to 1920.
First, we establish that the US has two primary backyards (outside of the continental US): 1. The Caribbean. 2. The Pacific. Any issues in the Pacific are superseded by issues in the Caribbean, which are only superseded by issues in the North American continent. This will not alter; the US will not trade territory in either of these for territory in Europe.
My proposed idea that gets you close within the timeline:
The US defeats the South handily after some slave states refuse to secede (say North Carolina, Tennessee, and Arkansas, leaving Virginia isolated). The war does not result in the gross death total that it did OTL (I think... 2% of the US population at the time? That's WW1 levels, and is a big part of the reason the US did not want to get involved in WW1 in the first place; they had fought a war on a similar scale, and wanted little of it). Slavery is not repealed during the war, but an amendment is passed soon after the war and slavery is ended, with the various southern states being forced to ratify the amendment to reenter (at least, they are until enough pass it that it becomes part of the Constitution). This spurs debate over what to do with the former slaves. As per OTL, talk of annexing Santo Domingo occurs, but fails as well. However, due to the US not being spent by the Civil War, the debate over making a black-majority state/territory continues, and the US turns to Liberia in the 1870s.
This occurs just as the scramble for Africa heats up. Liberia accepts US help, desiring more former slaves as citizens to strength the party rule in the nation, and it slowly become a protectorate. During the scramble, in exchange for the US pressing Liberian claims, they end up becoming a US territory. (the distance from North America means that autonomy is the order of the day, and is easily helped out by the ongoing immigration program the US is encouraging, along with investment). So, we have the US active in Africa with a slightly larger Liberia, perhaps. For the purposes, we are going to avoid the US taking any other purpose of the division of Africa.
This status quo remains until the US and Spain end up in a Spanish-American war in the 1880s/early 1890s (or 1898; I'm just spitballing right here). The US, in better shape than OTL, maintains a larger navy that has at least a few modern ships (this is also helped by the need to grant some protection to Liberia). This means the navy does not rot to the degree it did OTL. Whatever the circumstances might be, the US proceeds about the same in Cuba and the Pacific. However, the presence of the US in Africa means that they manage to also launch strikes into Spanish Equatorial Guinea and, later, towards the Canary Islands. A successful assault in those islands finally drives Spain to the peace table.
In these terms, the US seeks the same terms from Spain as it would OTL (I'm not going into the whole flip a coin aspect over Luzon vs all of the Philippines; that was a dream that decided it.) plus a few other concessions: Spanish Equatorial Africa and, in exchange for the return of the Canaries, Spanish Sahara and Ifni (if you want to throw in Micronesia somewhere, that is fine. Those are finer timeline details that would have to be shown). While Equatorial Guinea will be viewed as a second Liberia, the Western Sahara will be viewed as a headache by the US (see the US reaction the the American-Philippine War) and, instead of setting up an independent government as in the Philippines, would likely be ceded to Morocco (perennially friendly to the US) in exchange for recognition of its annexation of Ifni as a naval base and, perhaps, the establishment of an indefinite lease.
Here, the US seeks the lease so as to, essentially, make Morocco a protectorate, but reality would show that it would be a relatively minor port that would merely protect the approaches to Liberia/Equatorial Guinea while at the same time warding off European interests in Morocco. This would really start to alter things, then, as with the US reaching out both to the east and the west at the same time. This might be accepted by the European powers, mostly because Morocco was a point of contention and the US making it a protectorate keeps the French, Spanish, or Germans from doing so.
Ifni is the closes I think you can get, unless you finagle some way to get Tangiers to join Ifni (US protectorship endorsed by Morocco as being the only way to truly make Tangiers neutral to European affairs. You basically take the Swiss Approach: The French want it, the Germans want it, the Spanish want it, the American don't really care, so the answer is obvious!) Wilsonian Armenia is just a bridge too far for me; Wilson was alone on his various points, and the US had no desires to involve themselves in European affairs until much later (especially that far away from Europe and close to a disintegrating Russia). A strong Russia would have outright annexed Armenia and the Straits if it survived the war, and Britain and Italy were picking over the pieces.
I'm sure there are plenty of holes in my logic, but that gives you a general idea.
...I feel like I go over this quite subject a lot (US States that almost were/might have been). I am tempted to make a Beedok-style Worlda that can show all of the historical opportunities. Been a while since I've made one.
Of course, a really bad WW2 could lead to all/some of USS Corsica, Sardinia, Sicily, Crete, and Cyprus coming into existence in the American Fleet, but that is, again, outside the scope of the OP's desired answer. Another one could be direct US enforcement of a neutral city, but that seems pushing it by quite a bit unless you have AANW-style US. Granted, Trieste was an original member of NATO...