As pointed out by other commentators here and in similar threads, the key to Roosevelt becoming President again in 1913 is to win the Republican nomination. Roosevelt as the Republican nominee is not certain to win, since the Democrats were making gains and Taft supporters would be bitter, but he would have a good chance of winning a close election. During that period (really before the Great Depression), there seems to have been a hard ceiling on the Democratic nationwide vote percentage in the high 40s, though given the relatively robust Socialist vote in 1912 the winner of that election was going to be a nationwide popular vote plurality winner. I did an analysis of this in another comment thread, and in electoral vote terms it would come down to who carried New York, which would be really close (IOTL 41% Wilson 28% Taft 24% TR -mirroring the national results except the positions of Taft and Roosevelt are switched, and four years later New York would be a Hughes state).
Roosevelt winning as a third party candidate is really unlikely but not impossible -Wilson's nationwide popular vote margin over Roosevelt was 14%. However, it is unlikely Wilson can be pushed much lower than his OTL 41% nationwide popular vote percentage. Only Cox and Davis on the receiving end of the 1920s GOP landslides, after Wilson reneged on his promise and took the USA into World War I, got lower nationwide popular vote percentages in that era. The best way to do it is to manufacture a major and well publicized scandal surrounding Wilson that knocks his nationwide popular vote percentage into Cox territory, and more importantly swings half of Taft's voters over to Roosevelt to make sure of blocking Wilson. Another possibility is a situation where Taft in fact agrees to stand aside for Roosevelt, maybe he is promised nomination to the next Supreme Court vacancy, but the stand-pat faction of the party forces through the nomination of Taft specifically to spoil the election. Taft does not wage a campaign, but due to being on the ballot and there being an anti-TR faction among the Republicans, still gets a significant vote share, say 14%, just not enough to actually spoil the election.