South Carolina diving in June is far from cold - it was t-shirt/boardshorts diving for me in July.
Your options in Charleston are mostly
Charleston Scuba - on the south side of town or
Low Country Scuba if you're staying on the north side of town (Mt Pleasant area) We had a dive scheduled with Charleston but they cancelled due to conditions so the only SC diving I've done is off Myrtle Beach. It's not the clearest you've ever seen but most of the 1/2 day diving (they do some of the same dives from Charleston) consists of shallower wrecks unless you do the all-day trips out to the Gulf Stream wrecks which are deeper.
Charleston will want you to do a refresher b4 diving with them. It's usually every other Sunday but since you're only there the weekend, ask about scheduling something at night when you arrive. IDK anything about Low Country other than what's on their website.
There's quite a few 60' dives off Charleston though. Except for the ledges are all anchored by wreck(s). Not always the biggest wrecks but generally the boat ties off on them so you just follow the line down. Most of the shops use a line system from the back of the boat to the mooring - you're encouraged to stay on it while descending. There
won't be a Divemaster in the water - they only dive down to set/detach the mooring line. Charleston charges $200 more for a private DM. The boat rides out are over an hour also generally. The crew generally helps divers on/off the boat.
The Subway Cars are a cool dive - they're all dropped in basically the same area and most (all?) are upright and penetrable. Since you dove the Vandy, you'll have no problem. If you don't have a wetsuit, watch for jellies while surfacing, they're attracted to sunlight. Purge your octo to clear them away while surfacing.
Be sure to visit downtown, the SC Aquarium is worth a look. There's B&B's in a couple areas in historic old houses also for somewhere different to stay. And the old market -
Historic Charleston City Market, Charleston, SC is worth a look. This may be the best restaurant in Charleston -
Charleston Steakhouse Restaurant | Halls Chophophouse - possibly in SC. If you're at all military buffs both Fort Sumter and Patriots Point are options. One of the Fort Sumter tour boats docks is at Patriots Point - the other downtown. I think there's only one official tour company allowed out there. It's historical but I was personally
underwhelmed by Fort Sumter - they've kept it pretty original.
I haven't been there in a while so some of this may have changed.