Looking at what bases (except for prime detectives) are already included in the tour, it seems to me that up to 36, we need at least five divisors:
Five divisors: 16 (tess) Six divisors: 18 (dine), 20 (score), 28 (cadeff), 32 (twive), 44 (cadell), 52 (cadithe), 68 (cazote), 76 (cadax), 92 (caflore), 98 (seneff) Eight divisors: 24 (cadex), 30 (kinex) Nine divisors: 36 (exent)
And up to 64 we need at least seven (I'm planning a guest post for 64):
Seven divisors: 64 (twex) Eight divisors: 40 (kinoct), 42 (exeff), 54 (exove), 56 (sevoct) Ten divisors: 48 (exoct) Twelve divisors: 60 (shock)
And then up to 120 we need at least nine:
Nine divisors: 100 (kent) Ten divisors: 80 (octess), 112 (setess) Twelve divisors: 72 (octove), 84 (sezzen), 90 (novess), 96 (extess), 108 (catrine) Sixteen divisors: 120 (hund)
There have then been so many numbers with 12 divisors that to continue, we seem to need at least fifteen up to 240:
Fifteen divisors: 144 (zenent) Sixteen divisors: 168 (cadexeff), 210 (zeffchick), 216 (excue) Eighteen divisors: 180 (scorove) Twenty divisors: 240 (kinexoct)
and at least eighteen past that:
Eighteen divisors: 252 (zentress), 288 (notwive), 300 (zenquint) Twenty divisors: 336 (exsevoct) Twenty-four divisors: 360 (kinoctove), 864 (twivetrine), 3500 (kinsekent)
and 360 is so hard to beat that in the range past that we really need to match it:
Twenty-four divisors: 420 (exsevess), 480 (exodess, exoctess), 504 (sevoctove), 540 (exovess, scoretrine), 600 (cadexquint), 630 (senovess), 660 (exdessell), 672 (exeftess, twivetress)
before we finally beat it.
Twenty-eight divisors: 960 (zenodess), 2240 (kineftwex) Thirty divisors: 720 (octovess), 1008 (senotess) Thirty-two divisors: 840 (seveszen), 1080 (noveszen), 1890 (setriness), 1920 (cadexodess)
And now, only the actual largely composites seem to stay interesting, as the lanterns in Oschkar's staircase flicker out and leave us groping in the dark.
Thirty-six divisors: 1260 (scorsenove), 1440 (novestess), 2016 (senotwive), 2100 (kinexsevess), 2400 (kinexodess), 3840 (tweightrick) Forty divisors: 1680 (sechitess), 2160 (kintestrine), 16800 (kinsechitwive)
And finally, all the companions of kinsevoctove:
Forty-two divisors: 2880 (novestwive), 4800 (kinzenodess) Forty-eight divisors: 2520 (kinsevoctove), 3360 (sechitwive), 3780 (scorsetrine, exsenovess), 3960 (kinoctovell), 4200 (exsekent), 4320 (kintwivetrine), 4620 (exsevessell), 4680 (kinoctovithe), 5280 (exodessell), 5400 (kinoveszen), 5940 (scortrinell), 6120 (kinoctovote) Fifty-four divisors: 7200 (kinovestess) Fifty-six divisors: 8640 (kintwextrine)
before we exit to the roof of the skyscraper.
Sixty divisors: 5040 (sevoctovess, kinsenotess) Sixty-four divisors: 7560 (senoveszen, kineffoctrine), 21000 (kinexsekent) Seventy-two divisors: 10080 (kinsenotwive)
All these are just "gut feelings", of course, fuelled by possibly inappropriate extrapolation.