Preventing implicit information flows by dynamic program analysis requires coarse approximations that result in false positives, because a dynamic monitor sees only the executed trace of the program. One widely deployed method is the no-sensitive-upgrade check, which terminates a program whenever a variable's taint is upgraded (made more sensitive) due to a control dependence on tainted data. Although sound, this method is restrictive, e.g., it terminates the program even if the upgraded variable is never used subsequently. To counter this, Austin and Flanagan introduced the permissive-upgrade check, which allows a variable upgrade due to control dependence, but marks the variable "partially-leaked". The program is stopped later if it tries to use the partially-leaked variable. Permissive-upgrade handles the dead-variable assignment problem and remains sound. However, Austin and Flanagan develop permissive-upgrade only for a two-point (low-high) security lattice and indicate a generalization to pointwise products of such lattices. In this paper, we develop a non-trivial and non-obvious generalization of permissive-upgrade to arbitrary lattices. The key difficulty lies in finding a suitable notion of partial leaks that is both sound and permissive and in developing a suitable definition of memory equivalence that allows an inductive proof of soundness.