To be able to generate a list of candidates, one should know some common ways to produce different principal parts from a given stem.
To produce the present stem (the first principal part as you call it) one might add a nasal before the last consonant.
To produce the perfect stem one might reduplicate the first syllable, with possibility for vowel gradation.
To produce the perfect participle stem one might add a t, with possibility for assimilation in the preceding consonant.
Using these rules to the stem tag-, one obtains the stems tang-, tetig- and tact-.
You can find more possible ways to play with the stems in many Latin grammars, and I will not attempt to list them here.
Instead, let me try to detail the process I would take to find nancisci from nacti.
First, the perfect participle stem seems to be nact-.
There is no sign of reduplication (we couldn't remove a syllable anyway since there is only one — and reduplication should only appear in the active perfect stem anyway), so the present stem might be nac-.
It is possible that the t has caused some assimilation, so it could be nag- or nah- instead.
There could also be a nasal in the present stem, so also nanc- and nang- might be possible.
If you manage to conclude that one option is nanc-, you will find the verb in a dictionary; you won't need to guess the incohative -isc-.
To see the first principal form, know the rules to generate later principal forms and try to guess which ones may have been used.
If I see a perfect participle of a new verb, I will not know the first principal part for certain, but I can make a number of guesses.
And know your irregular verbs — they may also come with prefixes.