A fundamental task in multi-agent systems is to match n agents to n alternatives (e.g., resources or tasks). This is often done by eliciting agents' ordinal rankings over the alternatives rather than their exact numerical utilities. While this simplifies elicitation, the incomplete information leads to inefficiency, captured by a worst-case measure called distortion. Recent work shows that making just a few cardinal utility queries per agent can significantly improve the distortion, and in particular that O(√n) distortion is achievable with two queries per agent. We generalize this result by achieving O(n^(1/λ)) distortion with λ queries per agent, for any constant λ, which is optimal up to a constant factor given a previously known lower bound. We extend this finding to the general social choice problem of selecting one of m alternatives based on n agents' preferences, achieving O(min(n, m)^(1/λ)) distortion with λ queries per agent.