The three steals brought Hamilton's total to 131, only 14 from tying Vince Coleman's record of 145 set in 1983.
The Reds' top prospect put on another baserunning clinic, reaching twice on balls that did not get past the pitcher.
After grounding out in the first inning, Hamilton beat out a roller in front of the plate that was fielded by Lookouts catcher Gorman Erickson. He promptly swiped second and third and came home when Erickson's throw sailed past third baseman Luis Nunez.
"He definitely has a lot of flexibility," Blue Wahoos manager Jim Riggleman said of Hamilton's freedom to steal at will. "There are times where we have been down a few runs and Billy will get the green light in a situation where a runner usually wouldn't. Sometimes it comes back to hurt us and sometimes it pays off."
In the sixth, Hamilton reached when pitcher Matt Magill fielded his ground ball and chose to look Chad Rogers back to third base. Rogers, Pensacola's starting pitcher, had opened the inning with his first career triple.
Hamilton stole second, took third as Bryson Smith bounced into a fielder's choice and raced home on a passed ball.
It was Hamilton's fifth multi-steal game in August. He has 14 thefts this month, more than any other Southern League team.
With 23 games left in the regular season, the 21-year-old shortstop sits in third place on the all-time single-season list, trailing only Coleman and Donell Nixon (144).
"Of course, I think [the record] would mean a lot [to him]," Riggleman said. "You're talking about some greats of the game, like Vince Coleman. You're in an elite class when you're talking about stolen base records, getting into the conversation with Coleman, Rickey Henderson, players that have taken this success to the Major Leagues.
"It's a tough season. It's a great credit to Vince to have that record and a great credit to Billy in how he has gone after it."
Hamilton started the season at Class A Advanced Bakersfield, where he stole 104 bases and batted .323 in 82 games. In just under a month with the Blue Wahoos, he's hitting .263.
Rogers (2-0) won his second straight since he was promoted from Bakerfield, limiting the Lookouts to three hits while striking out six over six innings.