To expand slightly on Aaron's answer, the most common use of augmented triads in "classical" harmony (and harmonic languages based on that) is as a modification of a dominant chord.
That is, V-I can be modified where the fifth of the V chord rises chromatically to the third of the I chord, creating a V to V+ to I progression. Sometimes, depending on voice-leading and the particular progression, the V+ can appear without the preceding (normal major) V.
Many types of dominant chord can be used: one sometimes sees a V+ with a seventh or a V+ as a secondary dominant, etc.
That's the most standard functional use of an augmented triad, as a kind of substitute note in a dominant chord. Otherwise, as Aaron said, they mostly occur as passing chords with semitone motion.
I'm well aware that this chord might occur naturally in a minor (key)
progression that uses the harmonic minor scale.
Yeah, that's more theoretical than practical. It's something that theoretically falls out of the "harmonic minor scale" which isn't really a thing used to build stuff harmonically. III in minor is almost always major. The only time III+ would ever occur in standard "classical" progressions would be if the chord was actually V+/VI, i.e., a secondary dominant.
(FYI: A bit of a digression, but the "harmonic minor scale" is a theoretical construct originally used as a shortcut to find the most common notes for most chords in minor. But it wasn't originally intended as a practical reflection of the nuances of minor-key harmony. In the 20th century and in some folk musics, it came to be used as an actual scale as the basis for melodic and sometimes harmonic creativity. But III+ is simply not a "normal" chord in the minor mode, at least the traditional use of minor-key harmony. One could just as legitimately build an augmented triad on most other scale degrees.)