Have the 1920 Republican nomination go to Leonard Wood (a Theodore Roosevelt protege) or Hiram Johnson (a pro-regulation populist) instead of the laissez-faire Harding, and give the pro-tax and pro-central-planning Hoover a major role in the new administration (probably Treasury Secretary in place of Andrew Mellon).
Whereas Harding and Mellon responded to the 1920-1921 recession by rolling back high wartime tax rates (cutting the top rate from 77% to 24%) and implementing a deep austerity program to keep the federal government running a surplus to pay down the national debt, a Wood+Hoover or Johnson+Hoover team would more likely keep taxes high, use the additional revenue to fund expanded poor relief, and increase regulation and central planning to try to root out "inefficiency" in the private sector. If the depression winds up deeper than OTL, implementing a Social Security-like program, a WPA-like program, and a National Recovery Act-like program is not out of the question.
If the Democrats nominate John Davis in 1924 as per OTL, I could very easily see him running to Wood's or Johnson's right. Davis was a conservative Southern Democrat, who IOTL opposed much of FDR's New Deal and argued several of the court cases challenging the constitutionality of various aspects of the New Deal. As TTL nominee, he'd likely serve as a rally point for conservative opposition to the quasi-New Deal policies of our hypothetical Wood or Johnson administration.
There's a variety of ways things can proceed from that point. Probably the best option in terms of this AHC would be for Davis to win and start trying to roll back Wood/Johnson's policies, but only succeed in rolling things back a little. Davis gets reelected in 1928, then the stock market crashes and the economy goes south again around 1930 or 1931, and Hoover or another pro-government Republican successfully defeats Davis's VP in 1932 by blaming the renewed depression on Davis's economic policies.