the question 

1. First chord is C-major in second inversion (i.e., bass line plays G).
2. The middle chord: bass note is G#. The other two notes are B and D. So it's like a G-major chord except that I am using a G#.
3. Third chord is a-minor
Does that middle chord have a name?

A : The middle chord has the notes B D G# (later arrange the G# as bass or whatever you wish, but these are the chord notes)
Here are the notes of G# minor chord : G# (root) – B (minor 3rd) – D# (fifth)
The only difference between G#m and our middle chord is that the D# in G#minor chord is flattened to D to get the middle chord.
That is, the fifth of G# minor is flattened to get the new chord.

When a minor chord has its fifth flattened, we get the diminished chord – So the middle chord in question, with notes G# B D, is G#diminished chord.

Now here are two more chords (four note) which are derived from this middle chord
G# dim 7th : G# D B F (the interval between each of the four notes is a minor 3rd – equal division of the 12 note scale, which means this chord is a gateway to 4 different directions)
Edominant 7th (E7th) : G# D B E

V to i motion

E7 chord to Am is a V7 to i motion – the dominant seventh chord of the 5th chord (with respect to the root) moves to the root minor chord.

E7 chord to A major is a V7 to I motion.

Note that the chord is noted as major or minor by using capital I or small i respectively.

We have C and Am. Then we want to add one more chord in between. Then the first choice is to look for a V to i motion.

We are moving to Am. So look for the 5th of A. It is E. 

So we could change the progression to : |C E|Am|

E7 is the dominant seventh chord with root E which works well as the fifth chord which resolves to the root chord A.

So we could further change the progression to : |C E7|Am|

Above we saw that E7 is :G# dim chord plus the E note.

The middle chord in question could be E7 if you add the E note. It is derived from the use of V7 to i motion – moving from E7 to Am

The Jazz Theory Book by Mark Levine is an excellent book if you would like to learn more about reharmonizing.

Indian guitar, carnatic, rhythm and chords, finding chords, playing by ear etc(Click to see)
Free resources at musicianself.com/rlo