You are not going to be able to "mud jack" near a liner pool. You could destroy the walls and plumbing. In fact, mudjacking is almost completely obsolete these days. As someone else said, you likely have a plumbing leak unless there is absolutely no plumbing or overflow drains on that side of the pool. If it was closer to the build, I would lean towards it being a compaction issue. If you are in clay soil, its possible to get some sinkhole action with heavy rains after extreme drought. That is usually reserved for Texas and Colorado though as I doubt you get five or six month long droughts in MS.
If you isolate that its not a leak, it may be some old compaction issues rearing their ugly head. There are some techniques with subterranean polyurethene injections that can fill voids and compact soils, but it gets expensive quick and you run the risk of the foam finding its way into pool plumbing and then you are digging it up anyway. They would probably require SPT tests from a Geotech as well. By the time its all said and done, you might be close to the cost of ripping out and replacing the pool. Uretek is the only company I would trust near a pool... Not cheap.
I haven't messed with liner pools since building a few for a guy from church as a summer job in high school. But I have done a lot of remediation work (and turned even more down) for gunite pools experiencing settlement issues. I would lean towards hiring somebody to hand dig out the area (It was likely backfilled with a sandy fill and not compacted really well at construction.) Once dug out you can inspect plumbing and refill in 3-4" lifts compacted properly after leaks are repaired if present. I would drain the pool partially as the dig is taking place and watch out for ground water, you don't want to float your liner.
Other option is to hire a pool plumbing repair company to see if they can identify a leak. If not, just fill the depressions from the top through the flower bed and monitor. If you do not have a leak and the pool liner is not floating, its just a compaction issue and all of the rain might have done what the pool builder couldn't. After filling and leveling, then you can put in some drainage. Not french unless you discover tons of ground water. Surface drains and proper grading. French drains should be used to move subterranean groundwater... Surface drains and grading move rainfall/surface water.
Post pics if you can I would like to see.
