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.