We extend our recent results on propagation of semiclassical resolvent estimates through trapped sets when a priori polynomial resolvent bounds hold. Previously we obtained non-trapping estimates in trapping situations when the resolvent was sandwiched between cutoffs χ microlocally supported away from the trapping: ∥ χ R h ( E + i 0 ) χ ∥ = 𝒪 ( h - 1 ) , a microlocal version of a result of Burq and Cardoso-Vodev. We now allow one of the two cutoffs, χ ˜ , to be supported at the trapped set, giving ∥ χ R h ( E + i 0 ) χ ˜ ∥ = 𝒪 ( a ( h ) h - 1 ) when the a priori bound is ∥ χ ˜ R h ( E + i 0 ) χ ˜ ∥ = 𝒪 ( a ( h ) h - 1 ) .