Not quite, but you're on the right track. The three leading contenders (Lockheed, Thiokol, and United Technologies) all proposed segmented booster designs. Lockheed's were to be shipped by barge, in a vertical position, as opposed to the horizontal rail shipping used by Thiokol. (Not sure about UT.)
A fourth contender, Aerojet, proposed a single, monolithic booster, to be built at a new facility in South Florida and shipped up the intracoastal waterway by barge. But the additional cost of the new facility made that proposal less competitive, where the others assumed use of existing facilities.
[You're right; there's not much hard info about this online. I'm relying here on Malcolm McConnell's Challenger: A Major Malfunction (1988), which in turn relied heavily on Rogers Commission testimony.]
"Oh what wouldn't I give to be spat at in the face..." -- a prisoner in "Life of Brian"