The distinguishable cluster approximation proposed in Paper I [D. Kats and F. R. Manby, J. Chem. Phys. 139, 021102 (2013)] has shown intriguing abilities to accurately describe potential energy surfaces in various notoriously difficult cases. The question that still remained open is to what extend the accuracy and the stability of the method is due to the special choice of orbital-relaxation treatment. In this paper we introduce orbital relaxation in terms of Brueckner orbitals, orbital optimization, and projective singles into the distinguishable cluster approximation and investigate its importance in single- and multireference cases. All three resulting methods are able to cope with many multiple-bond breaking problems, but in some difficult cases where the Hartree-Fock orbitals seem to be entirely inadequate the orbital-optimized version turns out to be superior.