What would a 2nd full term of TR look like depending on foreign policy ?
From wikipedia on his successor, William Howard Taft:
Unlike his predecessor, Taft did not seek to arbitrate conflicts among the other
great powers. Taft avoided involvement in
international events such as the
Agadir Crisis, the
Italo-Turkish War, and the
First Balkan War. However, Taft did express support for the creation of an international arbitration tribunal and called for an international
arms reduction agreement.
[87]
So I think a second full term Theodore Roosevelt might not be as diplomatically shy toward Europe, and may offer mediation services in any of the crises and wars in the first sentence. However, those situations, not involving China and Pacific trade, may not as urgently involve US interests, and may not also be as 'ripe' for successful American mediation. Roosevelt would be uninterested in the arbitration tribunal and arms reduction agreements, thinking them naive.
He will likely be equally willing as Taft to intervene in Central America and the Caribbean to 'keep order' and reduce the chances of European or Japanese intervention.
He might not be as calm in terms of restraining border troops in reacting to spillover from the Mexican revolution.
One of the biggest differences could be in foreign policy. Whereas Taft sought to ensure the application of the 'Open Door' policy in all parts of China, including Manchuria, even as Russia and Japan were reaching ever more explicit agreements to divide the region into respective spheres of influence for themselves, excluding others, Roosevelt would be far more likely to accept that these spheres of influence would be a more realistic solution than actual Chinese administrative territorial integrity. Additionally, knowing that immigration restrictions on Japanese entry to America were fraying Japanese nerves, Roosevelt would be more sensitive to not make the Japanese feel that America was *also* trying to block Japanese expansion opportunities on the Asian continent as well as the American.