You do need a comma here. There is a systematic answer for this.
Whenever a sentence begins with an adverbial phrase, use a comma.
An adverbial phrase explains things like: Manner, timing, reason, purpose among others. In this case, you have an infinitive of purpose as an adverbial phrase.
In your case:
To further our understanding of this phenomenon, (Adverbial of purpose)
we conducted yet another experiment. (main clause)
Other similar examples:
To help him start his car, we pushed it until it started rolling.
To better understand what they were doing, we started spying on them.
These phrases use infinitives of purposes as adverbial phrases. It is much more common to place them at the end of a sentence like so:
We pushed his car until it started rolling to help him start it.
One might be tempted to think that it makes no difference, but it does. The reason one would put this at the beginning is to emphasize the purpose. Consider this context.
To help him start his car, we pushed it until it started rolling. To help him repair it, we called a mechanic. To convince him to leave, we offered him some of the stale food we had.
Also consider this phrasing in which the first purpose is logical, whereas the second oen is somewhat surprising:
We started spying on them to better understand what they were doing. To join them, we confessed that had been spying on them.