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[Display omitted] • comprehensive model for multi-color action spectroscopy of triatomic hydrogen ions. • simulations performed for both cryogenic ion traps and storage rings. • promising approach to bridge the gap toward highly excited states of H 3 + that were inaccessible by spectroscopy thus far. The H 3 + ion is the simplest polyatomic molecule. Apart from its fundamental role as a benchmark system for theoretical calculations, it has attracted a lot of attention as one of the main drivers of an active interstellar ion-neutral chemistry network. H 3 + spectroscopy has contributed greatly to astrochemistry, and experiment and theory have both developed tremendously throughout the years, following the ground-breaking work by Oka. To date, the highest-lying identified H 3 + transitions were obtained using an action spectroscopy technique, employing stored H 3 + ions in a cryogenic ion trap. This approach was pioneered by Schlemmer and Gerlich for N 2 + , and is widely used today – in one form or another – for spectroscopic studies of interstellar ions. In this work, we explore the possibility to extend the range for H 3 + action spectroscopy even further, into the regime above 20 000 cm−1, in order to bridge the gap to the rich pre-dissociation spectrum of states near the dissociation limit that was obtained by Carrington et al. and remains unassigned after more than 30 years. [ABSTRACT FROM AUTHOR]