I think this is the chart...
| ii V | I V vi |
| ii V | I III VI7 |
| ii V | I
...I'm not exactly sure of your barlines.
The part I V vi can be called a deceptive progression.
The part I-III-VI(7) I think you mean in F major the chord would be F A D7
Functionally I think you could just say I V/vi V7/ii. After the tonic chord it's like the beginning of a circle of fifths of dominants. Roots descending by 5th to ii. So then the chart could be analyzed like this...
| ii V | I V vi |
| ii V | I V/vi V7/ii |
| ii V | I
In F major the named chords are...
| Gm C | F C Dm |
| Gm C | F A D7 |
| Gm C | F
The last five chords are a long segment of the circle of fifths.
We can list out just the roots with Roman numerals and letters...
F Bb E A D G C F
I IV viio iii vi ii V I
The diatonic chord qualities are...
F Bb Edim Am Dm Gm C F
I IV viio iii vi ii V I
Excepting IV and viio it totally fine to modify the other chord to become either major triads or dominant seventh chords as long as the function as secondary dominants. I put the sevenths in parenthesis to show it is optional without changing the functional harmony. I also used both kinds of Roman numerals...
F Bb Edim A(7) D(7) G(7) C(7) F
I IV viio V/vi V/ii V/V V I
I IV viio III(7) VI(7) II(7) V(7) I
...you can use other combinations of minor chords versus major/dominant sevenths in that part of the circle of fifths from iii to I. The important thing is the roots descending by perfect fifths.
So, the final part of your piece plays the full circle of fifths using a few secondary domiants but omitting the IV and viio chords.