A key challenge in dynamic information flow analysis is handling implicit flows, where code conditional on a private variable updates a public variable x. The naive approach of upgrading x to private results in x being partially leaked, where its value contains private data but its label might remain public on an alternative execution (where the conditional update was not performed). Prior work proposed the no-sensitive-upgrade check, which handles implicit flows by prohibiting partially leaked data, but attempts to update a public variable from a private context causes execution to get stuck. To overcome this limitation, we develop a sound yet flexible permissive-upgrade strategy. To prevent information leaks, partially leaked data is permitted but carefully tracked to ensure that it is never totally leaked. This permissive-upgrade strategy is more flexible than the prior approaches such as the no-sensitive-upgrade check. Under the permissive-upgrade strategy, partially leaked data must be marked as private before being used in a conditional test, thereby ensuring that it is private for both the current execution as well as alternate execution paths. This paper also presents a dynamic analysis technique for inferring these privatization operations and inserting them into the program source code. The combination of these techniques allows more programs to run to completion, while still guaranteeing termination-insensitive non-interference in a purely dynamic manner.