Pairing-based cryptosystems are generally constructed using many functions such as pairing computation, arithmetic in finite fields, and arithmetic on elliptic curves. MapToPoint, which is a hashing algorithm onto an elliptic curve point, is one of the functions for constructing pairing-based cryptosystems. There are two MapToPoint algorithms on supersingular elliptic curves in characteristic three, which is used by η_T pairing. The first is computed by using a square root computation in F_3^m, and the computational cost of this algorithm is O(log m) multiplications in F_3^m. The second is computed by using an (m-1)×(m-1) matrix over F₃. It can be computed by O(1) multiplications in F_3^m. However, this algorithm needs the off-line memory to store about m F_3^m-elements. In this paper, we propose an efficient MapToPoint algorithm on the supersingular elliptic curves in characteristic three by using 1/3-trace over F_3^m. We propose 1/3-trace over F_3^m, which can compute solution x of x³-x=c by using no multiplication in F_3^m. The proposed algorithm is computed by O(1) multiplications in F_3^m, and it requires less than m F₃-elements to be stored in the off-line memory to efficiently compute trace over F_3^m. Moreover, in our software implementation of F_3⁵⁰⁹, the proposed MapToPoint algorithm is approximately 35% faster than the conventional MapToPoint algorithm using the square root computation on an AMD Opteron processor (2.2GHz).