Catalyzes the succinyl-CoA-dependent conversion of glutarate to glutaryl-CoA. Can use different dicarboxylic acids as CoA acceptors, the preferred ones are glutarate, succinate, adipate, and 3-hydroxymethylglutarate (By similarity). 3-hydroxy-3-methylglutarate + succinyl-CoA = (3S)-hydroxy-3-methylglutaryl-CoA + succinate Belongs to the CoA-transferase III family.