I don't understand your question the way you worded it. Contributions to an HSA and reimbursement for expenses have two completely different sets of rules.
Your eligibility to contribute to an HSA ends when you enroll in Medicare or any other insurance that is not a qualified HDHP. HSA eligibility is based on your coverage as of the first day of the month. If you were enrolled in Medicare on June 1, then you are eligible for the HSA for 5 months, and your contribution limit would be as follows:
If you have a single HDHP, your limit is $3650 divided by 12 x 5 months = $1521 plus $1000 catch up (for over age 50) divided by 12 x 5 months= $416 for a total of $1937.
If you have a family HDHP, your limit is $7300 divided by 12 x 5 months = $3042 plus $1000 catch up (for over age 50) divided by 12 x 5 months= $416 for a total of $3458.
You can contribute any time during the year, even after enroll in Medicare, but only up to your limit as calculated above.
Once you have money in an HSA, you can use it to reimburse any out of pocket medical expense for you, your spouse or your dependents, no matter what kind of insurance coverage you have.
