The approximate computation of twin-width has attracted significant attention already since the moment the parameter was introduced. A recently proposed approach (STACS 2024) towards obtaining a better understanding of this question is to consider the approximability of twin-width via fixed-parameter algorithms whose running time depends not on twin-width itself, but rather on parameters which impose stronger restrictions on the input graph. The first step that article made in this direction is to establish the fixed-parameter approximability of twin-width (with an additive error of 1) when the runtime parameter is the feedback edge number. Here, we make several new steps in this research direction and obtain: - An asymptotically tight bound between twin-width and the feedback edge number; - A significantly improved fixed-parameter approximation algorithm for twin-width under the same runtime parameter (i.e., the feedback edge number) which circumvents many of the technicalities of the original result and simultaneously avoids its formerly non-elementary runtime dependency; - An entirely new fixed-parameter approximation algorithm for twin-width when the runtime parameter is the vertex integrity of the graph.