Libertarianism is a conservative political philosophy despite it being sold in the USA as either centrist or even "liberal", i.e. left-wing. In the USA we have four competing political groups, the Progressive Republican, the Conservative Republican, the Progressive Democrat and the Conservative Democrat, and these are overlaid upon a basically conservative shifted political spectrum, Socialism here being regarded as really left and Liberalism being mainstream. The American debate is bifurcated between economics and social issues, it unites and divides across political lines. This seems to be how Libertarianism holds sway, it looks socially liberal and economically Liberal, in other words progressive and conservative simultaneously. Added to this are the traditional regional blocks, East Coast, Midwest, South and West.
Having the Republican Party embrace progressive policy after TR and have the GOP do better at recovering from the Depression using the full power of State intervention. Have the Solid south hold more sway and embrace State's rights more fully to devolve power away from Washington. Potentially civil rights would be the issue that drives this, so instead of the Dixiecrats coming over to the GOP, the Democrats gain a block of conservative moneyed voters/donors, the black and labor voter stays more fully GOP as they stay more Centre drifting Centre-left. This leaves the Democrats to hold the Centre-right, drifting right.
Perhaps having Hoover being more the New Deal progressive and the Democrats being in opposition, one needs the moderate East coast to have more sway longer, the Midwest conservatives stay opposed to the Southern conservatives, etc. I think the POD that might get you there could be TR runs in 1908 as his "second full term", Taft or another Republican follows orderly in 1912 so no "progressive" Wilson administration, the Dems are in power when the Depression occurs and take the blame, likely we have the GOP following a Democratic Presidency 1920 to 1928, in power as Depression occurs but can blame the Dems or the Dems are in power 1928 to 1932 or so, it is the GOP who offers the "New Deal". That is what I ponder when I toy with TR running in 1908 as opposed to handing over to Taft. 1912 becomes pivotal in a whole new way.