Your basic plan sounds workable. I assume your crawl space is adjacent to a full basement area? I have a couple of questions. Where does the existing drain tile system empty? Is there a difference in level between existing basement floor and footings of the crawl space? the answers to these questions may change my answer to you. But basically, adding a French drain or perimeter drain tile should help your situation. After your trench is dug, be sure to line it with several inches of gravel/crushed stone. Use perforated pipe with inlet holes or slots on only one half of the surface and be sure the slots are facing up when installed. Wrap the pipe in common landscape cloth before backfilling with crushed stone to prevent the drain pipe from filling with sediment. Lay another layer of cloth over the stone and complete the backfill with clean grave. Not sure how you plan to connect to the basement drain system, but that is another topic. This kind of work is always difficult, especially when working in a crawl space with limited head room. Good luck.
Options include looking for discoloration on surrounding areas and even putting a paper towel or Kleenex in suspect areas in order to detect a small leak that may not have been sufficient to create noticeable discoloration. Depending on the volume, you may be able to hear dripping in order to help trace the leak. And you may wish to reconsider whether the initial diagnosis was correct or whether the repair was sufficient.
Here are three causes that weren't obvious to us that you may wish to consider:
(1) Around the tub spigot: We had a situation in which we had a leak from a bathroom on the ceiling of the dining room beneath it. We re-caulked the tub and shower surround, did a small test, didn't see any water leaking, and thought we'd taken care of the problem. The next time someone showered the leak reappeared. We eventually figured out that the problem was around the spigot: Water from the shower was hitting the spigot, and because there was not a good seal around the spigot, water was running behind the spigot, down to the floor, and eventually into the dining room below.
(2) The actual copper supply line connections--if you're fortunate enough to have access to them without tearing out the wall: In another house, one with only one floor, we thought we had a leak in our shower pan, but during a remodeling discovered that the hot supply line had not been soldered correctly and water had been squirting out between the wall and eventually out through the brick in the adjoining exterior wall.
(3) Diversion due to insulation: We had another leak that we thought was due to a problem with the flashing between the roof and adjoining siding. After tearing up the roof and the ceiling of the room being damaged, we discovered that the problem a poorly-sealed nail in the roof itself. However, this location was several feet away from where water damage showed up. In this case, the insulation directly below the small leak was sufficient to keep the water from running straight down. Instead it wound around and into another room.
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For the crawl space I would not get a humidifier for this it will dry on its own. I would put a fan in the area to circulate the air and decrease the drying time if you have a fan but I would not purchase anything special to dry out the crawl space it will dry on its own.