也 lines a second item up as the same kind, flat alongside the first (he too / likewise). 還 piles the second on as an extra load, stressing there is more on top. Level equivalence vs added weight.
再 repeats or continues the same action a further time (do it again). 還 adds a different, second activity. Same act once more vs a new act stacked on.
不但…還…
The 不但…還… frame opens the count with 不但 (not only) and needs both halves. Additive 還 here stands alone in the second clause without 不但 leading the first.
他白天上班,晚上再當家教。 (再 = repeat the same act) → 他白天上班,晚上還當家教。 (還 adds a different second activity)
他白天上班,而且他太太晚上還當家教。 (new subject under 還) → 他白天上班,晚上還當家教。 (keep one subject for additive 還; a new subject takes 而且)
他不工作,晚上還當家教。 (first activity cancelled, nothing for 還 to stack on) → 他白天上班,晚上還當家教。 (the first activity must stand for 還 to add to it)
English 'also' or 'too' often drops to the end of the clause; Chinese 還 sits up front, right before the second verb, and signals an extra load rather than flat equivalence.