The Propose-Test-Release'' (PTR) framework is a classic recipe for designing differentially private (DP) algorithms that are data-adaptive, i.e., those that add less noise when the input dataset isnice''.We extend PTR to a more general setting by privately testing data-dependent privacy losses, rather than local sensitivity, hence making it applicable beyond the standard noise-adding mechanisms, e.g. to queries with unbounded or undefined sensitivity. We demonstrate the versatility of generalized PTR using linear regression as a case study.