Haskell, 58 bytes

d=['0'..'9']
f s=snd(span(<s)$mapM(\_->d++['a'..'z'])d)!!1

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A very brute-force strategy: generate all the length-10 base-36 strings in order, and find the one that comes after the input in the list. Take an enormous amount of time on strings far from the start of the list.

Haskell, 58 bytes

d=['0'..'9']
f s=snd(span(<s)$mapM(\_->d++['a'..'z'])d)!!1

Try it online!

A very brute-force strategy: generate all the length-10 base-36 strings in order, and find the one that comes after the input in the list. Take an enormous amount of time on strings far from the start of the list.

Haskell, 60 bytes

q '9'='a'
q c=succ c
f(h:t)|any(<'z')t=h:f t|r<-'0'<$t=q h:r

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Reads the string left to right until it reaches a character followed by a suffix of all z's, which may be empty. Increments that character, and replaces the z's with 0's.