Let fρ(L) indicate the smallest integer so that every curve on a fixed hyperbolic surface (S, ρ) of length at most L lifts to a simple curve on a cover of degree at most fρ(L). We provide linear lower bounds for fρ(L), improving a recent result of Gupta-Kapovich [6]. When (S, ρ) is without punctures, using work of Patel [9] and Lenzhen-Rafi-Tao [7] we conclude that fρ(L)/L grows like the recipr...