The only serious mentions of hockey I found in Chattanooga are of the inline skating variety.
Near as I can tell, the closest minor league teams to Chattanooga are currently the Knoxville Ice Bears (SPHL) and our Gladiators (ECHL).
While the city wouldn't be a bad place to put an AHL team, they'd definitely need a building that can support a team. The only arena I can find listed in Chattanooga is McKenzie Arena at the University of Tennessee's Chattanooga campus, and it seats ~10k for basketball.
I live outside of Chattanooga, and I'm utterly baffled as to why this town has absolutely zero hockey infrastructure. None. There isn't even an ice rink here; depending on which side of town you live in, if you want to play ice hockey (vs inline, purists), you've got to drive to Kennesaw, Knoxville, or Huntsville. I did this for years when it was still playing beer league, but that got to be far too taxing.
Chattanooga has an excellent outdoor/adventure sports infrastructure. Case in point, the Erlanger Marathon was yesterday and had close to 2000 total participants from across the country, and was very well attended by local spectators. But in the winter time, there is nothing much to do as a community/ civic entity. An SPHL franchise is a perfect entertainment product due to lack of competition, nearby teams in Huntsville and Knoxville, and lack of in-season competition.
Sadly, there is no arena here. Mckenzie is a dump, and has no ice plant. Plus it's wholly-owned by UTC. There's a massive push by local government to build another stadium for the Lookouts, our AA baseball team, but truthfully a new arena would be far more beneficial for the community as a whole, because of the nature of events it could/ would host.
Rant off. Tl;dr: Chattanooga is a perfect location for SPHL or ECHL, but lacks the necessary infrastructure.