You become HSA-eligible on the 1st day of the 1st month that you meet all HSA-eligibility criteria. This means if you have a qualifying HDHP that begins on March 15, you don’t become HSA-eligible until April 1.
Also, if you lose your HDHP (High Deductible Health Plan) coverage, you maintain HSA-eligibility until the end of the month in which you lose such coverage.
Source: IRC Sec. 223(c)(1)(A)(i)