Modified Look-Locker inversion recovery (MOLLI) is a commonly used approach for myocardial T1 mapping, and yields high precision as well as reproducibility. This technique requires one breathhold per slice, resulting in prolonged examination time when full left ventricular coverage is required. Our previously developed fast multi-slice myocardial T1 mapping (FAST1) technique showed potential for time-efficient full left ventricular coverage. In this work, the FAST1 sequence is further improved using Bloch-equation-based fitting for higher T1 accuracy and heart-rate independence. Compared to MOLLI, the proposed new FAST1 can yield higher T1 accuracy, enhanced tolerance to T2 and heart-rate variations, as well as similar repeatability and T1 map quality at 1.5T.