You're omitting a little detail: he would almost certainly have been president of the United States again! (Certainly Leonard Wood and Hiram Johnson would not have run if TR did in 1920; and many people who supported other GOP candidates might have supported him, too. His violent criticisms of Wilson made it easier for the Old Guard to forgive him for 1912.) This means that you cannot assume that everything else would go the same; certainly you can't assume that FDR (who only barely won the New York governorship in 1928 in OTL) would still become president.
This is fascinating in its own right. Before getting into the fascinating questions of what TR does in office in place of Harding and Coolidge, I'd like to skip ahead and put the cart before the horse to answer the OP's question.
Assume for a moment there is a great depression about on schedule and a Democrat wins and implements a new wave of progressive reform similar to the New Deal [I agree FDR's political success is unlikely here, so it would be somebody else], how would old TR react. I suspect he would be an outspoken, critical curmudgeon. However, the specifics of his critique would not match those of the New Deal's old-line conservative GOP detractors, no matter how much they might want to make tactical use of TR's criticism. Even if it didn't resonate with the public and elites, TR would probably bellyache about insufficient naval spending and if the Democrat did a "Good NEighbor" style retrenchment from intervention in the Caribbean, he would probably criticize that too.
---OK, let's get back to TR in office 1921-1925. How is he going to differ from Harding and Coolidge?
Will TR appoint Andrew Mellon to Treasury? Would he cut taxes as much as Harding? Will he be any less hostile to organized labor than Harding and Coolidge.
On internal racial issues I expect nothing would get done and in fact TR would probably avoid the gestures Harding made towards African-Americans (insisting on having mixed admission to his speaking events in the south).
I suspect immigration restrictionism would develop similarly to OTL.
How would TR be on anti-trust issues and Supreme Court appointments, and to what effect?
In international policy, who would he appoint as Secretary of State, and would he go for the Washington Treaties as constructed in OTL? In particular, might he have a big problem with the agreement to forego fortification of US bases in the western Pacific. Would he agree to negotiated arms limitation at all. No matter what approach he takes towards arms control treaties and Far Eastern issues, he would probably be disappointed by Congress in his hopes for the Naval budget and ship construction.
