The quantum phase estimation (QPE) algorithm is used to determine the phase of a quantum state and is one of the key subroutines for many quantum algorithms, including the HHL algorithm for solving linear systems of equations, quantum amplitude estimation, and quantum principal component analysis. In standard QPE, ancilla qubits are prepared in a uniform superposition state. In this paper, we generalize this notion and allow for an arbitrary initial ancilla state, which we refer to as a taper. In particular, for this modified version of QPE, called tapered quantum phase estimation (tQPE), we minimize the probability of outputting a phase estimate that deviates from the true phase by a certain amount in the worst-case and average error settings. Using phase-concentrated tapered estimates, we show that the number of extra qubits in tQPE required to guarantee that the output of the algorithm is $\delta$-close to the true phase with probability at least $1 – \epsilon$ can be reduced asymptotically to $m =O(\log\log(1/\epsilon))$, which we prove is optimal. We then show that the worst-case success probability for QPE (with no extra qubits, $m = 0$) can be improved by approximately $20\%$ using an optimal taper. Finally, we demonstrate that the mean success probability can be improved, relative to standard QPE, in all cases.