No, you can’t contribute into the HSA for 2021. Eligibility is determined based on your medical coverage on the first day of the month. If you did not have qualifying medical coverage on December 1, 2021, then you can’t make any contributions for 2021.
If, by some miracle, the medical coverage of your new job is retroactive to December 1, you could use the last month rule to contribute the full amount for 2021, which would be $3650 if you are enrolled in a single HDHP, and $7300 if you are enrolled in a family HDHP. To use the last month rule, you must be covered on December 1 and you must keep your coverage for the entirety of the following year (2022). If you lose your coverage during 2022, your last month rule contributions will be retroactively deemed ineligible and you will be assessed a penalty on your 2022 tax return.
