After two seasons with the Saints, he was released in a salary-cap move and quickly signed with the Dolphins.
"I left New England and went to New Orleans, and organization wise and everything it's a big difference," Jones said last week. "I don't want to get negative and everything, but it was a big difference. Leaving there and coming here is kind of like the organization and everything went back up."
At that point, Jones was asked if the Dolphins were more like the Patriots and he replied, "Same type of thing."
Of course, that has to sound good to Dolphins fans considering the Patriots' enormous success since 2001.
It also shouldn't come as a major surprise considering Saban coached with Bill Belichick in Cleveland in the early 1990s.
"He teaches a lot, I'll tell you," Jones said of Saban. "He's out there teaching new techniques, their demeanor is kind of the same. It's just the same as New England. It's a big difference from New Orleans, put it like that."
Jones is expected to start in a secondary that will feature three new starters, with cornerback Sam Madison the only holdover from last season.
But Jones isn't worried about the heavy turnover, partly because of his confidence in Saban.
"He's been here before," Jones said. "He's been with Belichick in Cleveland, he's been with Houston, so he knows everything about the NFL. He knows what it takes to go out there and win. I know the defense we're running now, when he was in college and when he was in the pros, it was No. 1-ranked, so it should be good."