grammar → 除了…以外,還/也…
TSUMUGU · TBCL 3 (est.) · 語法
除了…以外,還/也… grammar point · tier 2 · additive 除了…以外,還/也… (besides X, also Y)
Besides the named item, the count keeps going: set X aside, and there is still more.

字源 FORM what the parts do

除了 clears the named item off the step and sets it down as settled (了); 以外 walls it off as one already-counted side; 還/也 reaches back across the wall and adds what is still inside the count. The named item is parked, not subtracted.

故事 STORY a scene to remember it by

One item is set aside on the shelf; the hand goes back to the pile and pulls out more of the same.
自撰字源記憶法
框 · Frame
除了 [item set aside] 以外,[subj] 還/也 [more of the same]
觸 · Trigger
You name one thing, then want to add that there is more of the same kind beyond it.
序 · The move
1Put the already-known item inside 除了…以外.Is this item being set aside, not subtracted away?
2Open the second clause with 還 or 也.Without 還/也 it reads as 'except X' — exclusion. The resumption is what keeps the count going.
3Place the new item of the same kind after 還/也.Does the second item add to the first, rather than replace it?
例 · Examples
1這次的活動除了besides / apart from杯子以外outside (closes the bracket)also, in additionto give (as a gift)毛巾towel
Besides the cups, this event is also giving out towels.
界 · Boundary
除了…以外 alone (no 還/也)
With 還/也 the named item is set aside and MORE is added (additive: besides X, also Y). With 除…以外 standing alone, often closed by 都/只, the named item is the one EXCEPTION and everything else differs (exclusion: except for X). The 還/也 is the switch between the two readings.
連…都… (even)
連…都… pushes in the hardest case to imply the rest. 除了…以外,還/也… names one item, then adds another of the same kind, with no extreme and no implication.
✗ 除了杯子,送毛巾。 → ✓ 除了杯子以外,還送毛巾。 (close the bracket with 以外 and resume with 還/也)
✗ 除了杯子以外,送毛巾。 → ✓ 除了杯子以外,還送毛巾。 (drop 還/也 and it reads as exclusion, not addition)
✗ 除了我以外,有十幾位同學收到郵件。 → ✓ 除了我以外,還有十幾位同學收到郵件。 (the additive reading needs 還/也 to carry it)
English 'besides X' fronts the item with one word; learners stop at 除了X and omit 以外 and 還/也, which collapses the additive sense the frame is built to carry.