Based on the classical music I've listened to, you don't even need to insert any chords in between i and an augmented 6th chord.
Chords I have heard inserted between i and an augmented 6th chord and put immediately before the augmented 6th chord include V (J. C. Heed's "In Storm and Sunshine"), ♭II a.k.a. the Neapolitan (Franz Schubert's Moments Musicaux No. 5 in F Minor), and (♭)VI (Moments Musicaux No. 5 again).
I'd imagine that vii°6/5 of V would be a good chord to put between i and an augmented 6th chord, though, as it's one note away from a German Augmented 6th, and the root of vii°6/5 of V moves nicely down one semitone to the root of the German Augmented 6th (e.g., in F minor, D-F-A♭-B to D♭-F-A♭-B).
Often, there is more than one chord between i and an augmented 6th chord.