Thanks, Brandy. Usage sometimes departs from the explanations given in Adams, "Western Words," and other glossaries.
Larkin and Sandburg, who were the first to print the song, already differed (hoolian and hoolihan); both, I think guessing at the anon. composer's words (Lynn Riggs?).

Your definition is close to Adams for 'hoolihan,' which is to leap forward and alight on the horns of a steer in bulldogging and to knock it down (now barred at most rodeos).
Doing it by the tail is new to me.