Why is there a musty smell from my HVAC system even after coil cleaning?
Musty smells in the evaporator coils are common. You really need a flaming cleaner and the coils need to be cleaned 2-3 times to get everything out.
I use a product called Nu-brite  from My-Calgon it is a non acidic foaming coil cleaner, make sure your drain is working prior to use of it will make a mess. The foam pushes the dirt out.
What you are smelling is dust and grime when the system runs it collects moisture now that dust and grime are the “smelly socks” breeding ground you don’t smell it mid cycle because it is frozen at the start and end when the coils ate thawing you get the full odor.
1 cleaning usually helps 2 may do it but 3 has fixed even the worst (smokers) coils I have encountered with a clear water rinse after each cleaning.
Cool cleaning is suggested at least 1 time per year and with heavy use 2x.
Always have a filter in your system or this can start in just a few hours with dusty conditions.
Whatever you use don’t use an acid based cleaner acids damage the aluminum fins and shorten the coil life.