Partially-functional broadside tests attempt to maintain close-to-functional operation conditions by using scan-in states that are referred to as partially-reachable. An earlier procedure starts from the fully-unspecified state, and uses fully-specified primary input vectors to compute partially-specified states that are partially-reachable. The procedure is divergent in that the number of states it considers increases with every iteration. This paper describes a new, convergent procedure that starts from a subset of reachable states, and uses the fully-unspecified primary input vector to compute partially-reachable states. The procedure typically converges without any additional constraints. By starting from reachable states, the convergent procedure captures functional constraints that cannot be obtained by the divergent procedure starting from the fully-unspecified state. Experimental results for benchmark circuits demonstrate the implications on the generation of partially-functional broadside tests.