I assume that your progression is, for example Ab - Cm - Bb - Cm - Fm - Cm - Bb - Cm in C minor, or F - Am - G - Am - Dm - Am - G - Am in A minor.
Chord progressions are harmonic swaying from side to side, and back to center. The tonic is your center position, center of balance.
One way to think about chord progressions around a tonic is what I call 3 + 3 basic chords: 3 on the major side and 3 on the minor side. Let's say the major side is "front" and minor side is "back", dominant is "left", tonic is "center" and subdominant is "right". We have two dimensions: back-to-front and left-to-right.
If we map your chord progression F - Am - G - Am - Dm - Am - G - Am to this model, we get the following dance moves for the left-right dimension:
Right - Center - Left - Center - Right - Center - Left - Center
Now to your original question: how can a progression start with something else than a tonic? If we map that to the dance moves chart, the question becomes: how can a sequence begin from anywhere else than center position?
This is about the art of song-writing IMO. What kind of a harmonic story do you want to tell? Where do you place the moments of feeling tense, and where do you place the moments of release, in your story? It's a matter of taste, but your chord progression could work as a chorus or maybe a bridge. Or why not a verse as well, as long as you have established where the center position is, with an intro or something.
To make the progression slightly more interesting - and this is a matter of taste - you could add some variety in the back-to-front direction. How about: F - Am - G - Am - Dm - C - G - Am ? The C in there makes it slightly less boring, when it's not going back home to Am all the time, immediately. And if you want some extra interest and glue, how about repeating it but with a special interest chord in the middle... and add more back-to-front movement by replacing the G with Em which is also on the dominant side:
F - Am - G - Am - Dm - C - G - Asus4 ...
F - Am - G - Am - Dm - C - Em - Am
What I'm saying is, it's about form and structure. What is your big structure. Is it a limerick, haiku or 12-bar blues? What is the form.