The problem of whether and how one can compute the twin-width of a graph -- along with an accompanying contraction sequence -- lies at the forefront of the area of algorithmic model theory. While significant effort has been aimed at obtaining a fixed-parameter approximation for the problem when parameterized by twin-width, here we approach the question from a different perspective and consider whether one can obtain (near-)optimal contraction sequences under a larger parameterization, notably the feedback edge number $k$. As our main contributions, under this parameterization we obtain (1) a linear bikernel for the problem of either computing a $2$-contraction sequence or determining that none exists and (2) an approximate fixed-parameter algorithm which computes an $\ell$-contraction sequence (for an arbitrary specified $\ell$) or determines that the twin-width of the input graph is at least $\ell$. These algorithmic results rely on newly obtained insights into the structure of optimal contraction sequences, and as a byproduct of these we also slightly tighten the bound on the twin-width of graphs with small feedback edge number.