Swift began his round with three consecutive birdies and Talbert Griffin ended with a 1-under 71 to give Alabama a second-round total of 7-under-par 141 at the Fazio Course at the Club at Carlton Woods.
Texas, which led after the first round, fell into a tie with South Carolina, two shots back.
Swift, who along with Alabama's third team member, Steve Hudson, was on the state's 2005 runner-up team, has a sparkling two-day total of 137. Still, Swift thinks there is room for improvement.
'I think I can play a little better,' Swift said. 'I left a lot of putts out there.'
The best two scores of each state's three players are counted in each of three days of the competition, which concludes Thursday. Hudson had a 1-over-73 Wednesday, meaning his effort wasn't counted toward Alabama's total for the day.
The Texas team of Trip Kuehne, Jonathan Mathias and Terrence Miskell combined for a 146 on Wednesday. Kuehne particularly struggled, going 5-over during his final 10 holes, finishing with a 74. Miskell had the same score, while Mathias finished at even-par.
The South Carolina team, which was behind by eight shots after round one, continues to push for its first title. Steve Liebler led the effort Wednesday with a 68, while Rick Cloninger shot a 2-under 70.
Georgia is alone in fourth at minus-4, while Tennessee, the 2003 champ, is tied with Maryland at 3-under.
There are 52 teams from all 50 states, Puerto Rico and the District of Columbia entered in this event, which takes place every other year.