If I'm not mistaken, Hoover became more conservative as the 1920s progressed. As a businessman/engineer, one suspects he'd have been alarmed at the freewheeling practices on Wall Street in the earlier years of the decade, and would have decided to apply some sort of solution as a result (perhaps an indirect one) of the Panic of 1923, assuming it happens as in OTL. Perhaps he'd have fought for some sort of regulatory agency; i.e., a proto-SEC. I could see Charles Dawes (perhaps in his role as Secretary of the Treasury?) as the guiding light for this sort of move. It wouldn't prevent the crash of 1929 entirely, but I suspect it would go a long way toward mitigating it: perhaps now we'd speak of "the recession/panic of 1929" instead of "the crash".
It wouldn't surprise me if Coolidge were Hoover's running mate (ye gods, what a dull, colorless ticket!) given his popularity after the Boston police strike. I don't see Coolidge as the 1928 nominee, though; that wasn't in vogue then for sitting vice-presidents to get much consideration for the top job unless they fell into it through a presidential death. Guessing Lowden or Sproul would be the '28 GOP nominee.
The one thing I'm unsure of is what Hoover would do about the Klan and its bald grabs at vigilante power, especially in cities like Dallas and entire states like Indiana. He wouldn't have any time for the Klan, but would he push for legislation to rein it in once and for all? That may be worth a thread in its own right.
I mentioned Dawes as Hoover's SecTreas. Chances are Charles Evans Hughes would get State, Philander Knox would get Justice, and maybe Hiram Johnson would get the War Department. To be sure, there would be nothing approaching Teapot Dome: Hoover was dull as beige paint but rigorously honest, including his choices of associates. Side note: expect former president Taft to get appointed to the Supreme Court as in OTL.
Who opposes Hoover in 1924? Guessing you may get a ticket with McAdoo at the top and maybe Oscar Underwood as his running mate (sort of recognition of one of the few progressive thinkers in the south at that point). The election would be closer than in OTL but you'd still have a Hoover win.
In 1928 I see no reason why Smith wouldn't be the Democrats' nominee. His opposition, either Lowden or Sproul as noted above, would likely win, and perhaps carry a few border south states where the Bible Belt voted its prejudices.
Someone else take it from here.