Water – Geotextile Filter for French drain woven or non-woven

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.

Somewhat up to you, but standard practice (IMPE) is to cover the pipe with some rock, then fold the fabric over to "box" the pipe (overlapping the fabric, unless you want to get fanatical and seam it), then cover with more rock to keep it in place.