Not recommended. Too much time on either side of high tide the current is ripping under both the east and west span. Really not swim-able. You can stay out of the current by keeping to the sandy area in front of the beach but vis will be 2 ft or less. I've seen it go from literally six inches a couple hours before high to 30+ ft once the tide starts to turn. And then drop again just as fast.
Mntlblok should weigh in on this. He usually gets in 3 hours before high whenever he's in town. And I think has also dived other random times. Oh Kevin!!!
I *have* experimented some with other-than-slack-high diving at the bridge. While I have experienced some very poor viz at *all* tides, I *have* found diveable viz on a couple of low tide outings. I can also recall doing some snorkeling well after slack high with no major viz issues - primarily in the lifeguard's "swim" area. "Where" you go has a huge effect on the amount of current you'll experience. My sense is that the east side off the beach is more of a problem when the current is running, especially if you get over near the little bridge. Under that bridge, when the current is really ripping, even if you have some way to secure yerself to the bottom, it wants to try to rip yer mask off. And, it's really dark and scary under there, anyway. I seem to go through my air more quickly under those circumstances.
If you go straight out south off the beach area, west of the swim area, the deeper it gets - and the closer you get to the channel - the less the current, even three hours prior to high tide. I find it aggravating to have a bunch of air left and have the tide change after slack high and then no longer be able to use that air as nicely as I could have if I had gotten in sooner. That area has the shopping carts, a couple of little wrecks, part of the snorkel trail and its rocks, and a few other bits of cover. The more "easterly" part of this area tends to be pretty much just sand - as opposed to the "broken shell"-type bottom - but also is typically covered with a rather deep layer of "silt". In my expereience, one doesn't even need to "touch" this layer in order to have it silt yer area up. Even a fin kick a foot away from the bottom can nasty it up pretty quickly. Of course, there tends to be enough current to clear it up reasonably quickly, and at three hours before high tide, you almost never have anybody *see* how badly you screwed things up.
I've found some interesting stuff off the beaches at times, but the eastern portion has stricken me as being "more" barren than the western part. Some of my "rarest" finds have been out in this western, sandy area. Some of the depressions with some flotsam lying at the bottom of them can yield some surprises. And, just hanging still for a minute or two and looking around in all directions can bring the occasional surprise. Some kind of sharp(ish) spike or stick or rod stabbed into the sand can allow one to hold still and let his fins dangle down current. I like to attach that stabber to the bottom of my camera housing. And, if yer doing macro photography, you can still have a pretty good dive even when the viz is down toward the lower end of acceptible.
I've also done a couple of dives around slack low. Definitely not as good viz - and sometimes horrible - but it has also been sort of OK sometimes, too. Were I less lazy, I'd probably try to dive both "slacks" on most every dive day I have there.