Two things you need to know.
First, the 6 month lookback does not always apply. If you apply for Medicare within 3 months before to 3 months after your 65th birthday, your enrollment will be effective on the first day of the month you turn 65. (If you were born on the 1st of the month, your enrollment is effective the month before.) If you enroll after your initial eligibility period, your enrollment date will be 6 months before you apply. So the 6 month rule only affects your HSA eligibility if you apply for Medicare late.
Then,
Your Medicare date determines how much you can contribute to an HSA, but not when. For example, if you are covered by a family HDHP in 2024 and you are over age 55, your contribution limit is $9300, that's $775 per month. Suppose your Medicare enrollment is effective May 1, 2024. That means you were eligible to contribute to an HSA for 4 months, or $3100. That's your contribution limit for all of 2024, but it doesn't matter when in 2024 the contributions are made. You could contribute $200 per month, even after enrollment, and you would be fine because the total contributions are less than your eligible limit. But if you contributed $600 per month for the whole year, then you would have an excess contribution of $4100.
The tax consequence of an excess contribution is that you don't get a tax deduction, AND you pay a 6% penalty in the year you make the contribution, AND you pay another 6% in every future year that the excess contribution remains in the account. You can remove the excess contribution by making a withdrawal, not for medical expenses, and paying the income tax on the withdrawal. (Of course, you can also remove all the excess without tax penalty as long as you remove it before the tax filing deadline of the year you made the contributions, such as April 15, 2025 for contributions made in 2024.)
