Colombia would have had to agree to some kind of sale, rather than refuse all access, prompting TR to support Panamanian separatists to gain a client state for the canal.
Of course, TTL Jimmy Carter might well make the Zone and independent country, judging by OTL.
The degree to which it becomes a state largely depends on the extent to which US posessions acquired in New Imperialism become view as avenues for settlement/statehood (except Hawaii). Most of the problem was that these areas weren't seen as potential settlement sites since the frontier is gone. Add to that the idea that "the Constitution doesn't follow the flag" which allow the prosecution of an anti-guerilla campaign in the Philippines that produces enough resentment to mitigate any positives of statehood. Something like a wider US Zone might produce such effects, because it would create at least one trade city under US jurisdiction. Companies might relocate there, or at least have substanital presences. A Canal State (Roosevelt?) might encourage Puerto Rico to make up its mind and maybe the Philipinnes can have a referendum in the 1940s. Cuba is probably a lost cause, but perhaps a Caribbean state if US states in the Caribbean leads to some kind of purchase of the remaining European enclaves there say as part of the Marshall plan.
None of this is too likely since most people south of the Rio Grande and Miami tend to view the US as "Yanqui Imperialists" who aren't too much better than ones of the European variety.