In this paper, we propose several improved uniqueness conditions of solution for multilinear PageRank with order-3 irreducible stochastic tensors. By considering upper and lower bounds of solutions, the refined uniqueness conditions are more effective than the existing ones. Furthermore, as an application, sharp perturbation bounds for multilinear PageRank are also developed basing on the new uniqueness conditions. Numerical examples are given to verify the efficiency of the proposed uniqueness conditions.