"he" is correct, but "him" is more usual in speech (in my part of the globe).

I'm likely from that same part of the globe.

The form you're suggesting may even be regarded as acceptable, but as you say - "he" is correct. If you add the implied 'does' at the end of the sentence" it becomes:

"You swim much better than him does." Not good, right?

I think adding additional sentences to show alternate forms of expression is both acceptable and desirable, so feel free to add the form you're suggesting. Relaxed speech allows relaxed rules, right?

I'm American and I've heard it both ways. I looked it up out of curiosity, since I had thought that the form I used was the correct form, wrong! Here is what "Grammar-Monster" says -

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"John is taller than me" and "John is taller than I" are both correct. However, lots of your readers are likely to think that "John is taller than me" is wrong (even though it sounds natural), and a fair proportion of your readers are likely to think that "John is taller than I" sounds pretentious. The safest option is to expand the sentence after the than. For example:

"John is taller than I am."

This construction will satisfy all your readers and remove any ambiguity (more on that below).
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See 'http://www.grammar-monster.com/...an_he_him.htm' for further discussion.

Given this, I'll change my text.

Related:

[#3678268] Tom swims better than Mary. (CK) *audio*
[#3678269] Tom swims better than Mary does. (CK) *audio*
[#3502387] Tom swims better than I do. (CK)