Complexity of voting manipulation is a prominent research topic in computational social choice. The voting manipulation literature usually assumes that the manipulator is only concerned with improving the outcome of the election from her perspective. However, in practice, the manipulator may also be reluctant to lie, i.e., she may have a preference for submitting a vote that does not deviate too much from her true ranking of the candidates. In this paper, we study the complexity of finding a manipulative vote that achieves the manipulator's goal yet is as close as possible to her true preference order. We analyze this problem for three natural notions of closeness, namely, swap distance, footrule distance, and maximum displacement distance, and a variety of voting rules, such as scoring rules, Bucklin, Copeland, and Maximin. For all three distances, we obtain polynomial-time algorithms for all scoring rules and Bucklin and hardness results for Copeland and Maximin.