TAPIR is a highly accurate, precise and efficient method for T1 mapping of the brain. It combines an efficient slice-interleaving Look-Locker read-out to sample T1 relaxation by acquiring multiple k-space lines in one shot. However, mapping rapidly relaxing tissue requires the number of lines read in one shot to be small, thus increasing total measurement time. In this work we show how incorporating an interleaved spiral read-out into TAPIR enhances its T1 fitting abilities for rapidly relaxing tissue such as white matter myelin. Scanning time can be decreased by factors of up to 3.3 in comparison to classical Cartesian TAPIR.