Is this a new pool or one you just opened...if you just opened it what was the CYA on closing? The reason I ask this is that there are bacteria that can eat CYA in a closed pool and "poop" ammonia compounds which create a HUGH chlorine demand. If this is the case you might go through a lot of chlorine before it holds even with adequite CYA. You are not specifying whether the chlorine readings you get the next day are FC or TC. If they are FC I would be interested in what your TC readings are. If they are high then so is your chlorine demand.

Your initial readings of 0.1 ppm FC and 0.2 ppm TC look like pool store readings since drop based kits don't give numbers like that. (Your numbers look like they came from the LaMotte Waterlink Express or similar colorimeter)

What are you testing your water with at home to get your chlorine readings? OTO, DPD, FAS-DPD, Test Strips? (If you don't know then just let us know what test kit you are using.)

This info might help the folks on here get a better handle on your problem.