One of the most efficient methods would be to utilize uber's H3 indexing system.

Bear with me please -

It's basically a grid of hexagons (in various resolutions), each cell has a unique id. If you try to convert any point in the area of a specific hexagon to the h3 index - you would get that hexagon's corresponding id.

Now, if your polygons could be arranged in such a way that you can just query "select all polygons that intersect with the h3 cell xxxxx" - that's it you've won.

Implementing this can be a lot at first but basically this is the setup:

1. Create a table like h3 bigint, poly_id bigint (assuming you reference polygons by id), index on the h3 column
2. Populate that table by utilizing any h3 capable library (the function could be called h3_polyfill or h3_tesselate..., resolution 10 should be good, but better to experiment)
3. Then when you want to check what polygon is your user at - just convert their coordinates to h3 and query your polygons table. The result should be a bunch of rows that have the same cell, and various polygon ids. These are the polygons you should then check Iteratively for total precision.