An EASA fATPL is a non-licence of course, more a concept or collective noun. It is the collection of:-
Passes in ATPL writtens
CPL
ME/IR
MCC
Your CPL will have embedded PPL privileges, and so long as you did a skill test at some point in SEP, you'll be able to fly SEP in EASAland.
Most people do the CPL in a complex single, then ME/IR separately, and that route will ensure you can fly privately.
Certainly here in Britain you can fly an SEP on an FAA PPL(SEL) anyhow, but other countries may differ.