In Brandstädt and Giakoumakis 2, we reduce the Maximum Weight Independent Set (MWIS) problem on hole- and co-chair-free graphs to a corresponding problem on prime atoms. Here we refine these results by investigating atoms of prime graphs (avoiding the requirement that the graphs are prime and atoms) and also observe that the MWIS problem is solvable in polynomial time on (odd-hole, co-chair)-free graphs. •The Maximum Weight Independent Set (MWIS) problem on graphs is a frequently studied problem.•Clique separators and modular decomposition are powerful tools for solving this problem on various graph classes.•We combine these tools for solving MWIS efficiently on odd-hole- and co-chair-free graphs by showing that atoms of prime odd-hole- and co-chair-free graphs are nearly perfect and atoms of prime hole- and co-chair-free graphs are nearly weakly chordal.