In this article, we address the problem of computing all locally optimal solutions of a given multiobjective problem whose images are sufficiently close to the Pareto front. Such \epsilon -locally optimal solutions are particularly interesting in the context of multiobjective multimodal optimization (MMO). To accomplish this task, we first define a new set of interest, L _Q,є, that is strongly related to the recently proposed set of \epsilon -acceptable solutions. Next, we propose a new unbounded archiver, ArchiveUpdateL _Q,є , aiming to capture L _Q,є in the limit. This archiver can in principle be used in combination with any multiobjective evolutionary algorithm (MOEA). Further, we equip numerous MOEAs with ArchiveUpdateL _Q,є , investigate their performances across several benchmark functions, and compare the enhanced MOEAs with their archive-free counterparts. For our experiments, we utilize the well-established metrics HV, IGDX, and Δ _p. Additionally, we propose and use a new performance indicator, I _EDR , which results in comparable performances but which is applicable to problems defined in higher dimensions (in particular in decision variable space).