The Nominal Logistic and Ordinal Logistic platforms in JMP do not provide multiple comparisons. You would need to perform a lot of manual calculation after JMP.
The Generalized Linear Model platform in JMP provides Custom Tests. You can specify more than one test and an additional omnibus test is added over the set of all tests.
The Generalized Regression platform in JMP Pro does provide a variety of multiple comparison methods, including Tukey.