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You can say

1) Do you have more classes today?

2) Do you have any classes today?

3) Do you have other classes today?

4) Do you have any more classes today?

5) Do you have any other classes today?

6) Do you have another class today? (Thank you @Simon F)

7) Do you have (any) further classes/class today?

8) Do you have (any) additional classes/class today? ( Thank you @Mr Lister)