The jump rate and the jump-length probability distribution (JLPD) are calculated in a periodic potential with exponentially decaying memory friction, solving the generalized Langevin equation (GLE) by the matrix-continued-fraction method (MCFM). It is shown that the jump rate, as a function of the memory decay parameter, presents a turnover point; below the turnover a significant percentage of long jumps appears even at sufficiently high static friction, where long jumps are strictly forbidden in the absence of memory.