No, the alternative minimum tax (AMT) does not apply if the supplemental minimum tax is lower than the regular income tax. Here is an example for explanation.

After the exemption, David has a minimum taxable income of $170,100 in his alternative minimum. The regular income tax he pays is $55,195. What is the alternative minimum tax that David has?

Because the tentative minimum tax is lower than the regular income tax, there is no AMT payable. AMT is any tax that is greater than the tentative minimum tax than the regular income tax for the year.

David's tentative minimum tax of $44,226 ($170,000 x 26%) is less than his regular income tax of $55,195,  so that there is no AMT.