Yeah, just east of the existing P'Cola Beach Pier is the old pier rubble piled up. Its a fun dive with a variety of life. Vis can vary, but generally 15' or so when conditions are OK. It can get much better, and much much worse.
The road to Fort Pickens was knocked out by the hurricanes and they are still trying to figure out what they need to do in order to reinstall access to the Fort, much less start construction on whatever it is they plan on doing. There are jetties, Catherine(Norwegian bark), and Sport(tug on the bayside) off Fort Pickens, so it really sucks not having that access anymore...
Heading East...
Navarre Beach Pier is fun, lots of life here as well.
There is a sailboat off the bayside of Okaloosa Island's National Seashore Park that can be dived. Its small and hardly worth a full tank, but fun.
Destin Jetties are great, full of fish, a wide variety of them as well. There's even the chance of Manatees and Dolphins showing up - albeit slim.
Crystal Beach Pier(Destin) is uber shallow, probably better off snorkeling, but people have dived it as well.
St. Andrews Jetties in Panama City Beach are fantastic, lots of life, lots of coral/sponge, lots of colors.
Springs... Morrison's been noted, Vortex has been noted, Ponce De Leone Spring is within a park that doesn't allow diving. Cypress Springs is awesome and will likely be the clearest water you ever see, but requires a boat to access it(john boat, canoe, kayak, whatever).
Heading West...
Alabama Point Jetties at Perdido Pass in Orange Beach, AL, great little dive, lots of life at times. Never know what'll show up.
Gulf State Pier(Gulf Shores), its a long walk from the nearest public parking, but it can be a fun dive with lots of life.
Whiskey Wreck(Gulf Shores), great access, showers, restraunt onsite, lots of life, ect.
There are more shore accessible wrecks off Gulf Shores and Fort Morgan, but it would take serious bribing to get their locations.