The enhanced competition paradigm is an attempt at bridging the gap between simple and optimal auctions. In this line of work, given an auction setting with $m$ items and $n$ bidders, the goal is to find the smallest $n' \geq n$ such that selling the items to $n'$ bidders through a simple auction generates (almost) the same revenue as the optimal auction. Recently, Feldman, Friedler, and Rubinstein [EC, 2018] showed that an arbitrarily large constant fraction of the optimal revenue from selling $m$ items to a single bidder can be obtained via simple auctions with a constant number of bidders. However, their techniques break down even for two bidders, and can only show a bound of $n' = n \cdot O(\log \frac{m}{n})$. Our main result is that $n' = O(n)$ bidders suffice for all values of $m$ and $n$. That is, we show that, for all $m$ and $n$, an arbitrarily large constant fraction of the optimal revenue from selling $m$ items to $n$ bidders can be obtained via simple auctions with $O(n)$ bidders. Moreover, when the items are regular, we can achieve the same result through auctions that are prior-independent, {\em i.e.}, they do not depend on the distribution from which the bidders' valuations are sampled.