A problem which historically was first solved for primes, and then significant additional work was needed to prove the result for all integers.

The existence of infinitely many Ramanujan graphs of degree $n+1$ was proven for prime $n$ by Lubotzky, Philips, and Sarnak and for arbitrary $n$ 27 years later by Marcus, Spielman, and Srivastava.

This didn't just require additional work - the method was completely different, with virtually no overlap in ideas.