Wilson won 435 electoral votes to TR's 88
That does not constitute a 'strong second place'.
Wilson won 6,296,000 popular votes: (52% more than TR's 4,123,000).
Yes Roosevelt finished well ahead of Taft, but was nowhere near a "strong second" to Wilson.
TR's big problem was his impulsiveness, and since he had pledged not to seek another term, (on the eve of his 1905 inauguration, IIRC) he had to invent a rivalry with Taft to run again (and maintain his word by forming the Bull Moose party).
Say, that brings to mind the election of '92, H. Ross Perot and the election of Slick Willie.
History is indeed a fine teacher, what!!
And Perot did not cause the election of Slick Willie in '92. I was in the PhD program at American when that election took place. That precise question was addresses by several researchers, with different agendas. All came to this conclusion:
Perot's voters came equally from Clinton and Bush supporters, plus 2% of the total electorate who turned out to vote for Perot, but otherwise would not have voted at all.
You've got to get up early and pack a lunch if you want to play gotcha with this historian. I repeat, history is a great teacher. Take some more lessons.
John / Billybob