Double check your answers to the health insurance question. It sounds like Turbotax thinks you only qualified for 4 months, not 7.
To make tax-deductible contributions to an HSA, you must be enrolled in an HSA-eligible HDHP plan on the first of the month. If you were hired in mid-May, then the earliest you are eligible is June 1, assume your insurance coverage started on or before June 1. That gives you a standard eligibility of $2041.67.
However, under the last month rule, you are eligible to contribute the maximum $3500 for 2019 as long as you (a) are enrolled in HSA-eligible insurance on 12/1/2019, and (b) maintain your HSA eligibility through all of 2020. (If you contribute more than $2041 by taking advantage of the last month rule, then lose your eligibility during 2020, there will be an adjustment and penalty on your 2020 tax return.)
So for 2019, you can contribute $2041.67 without relying on the last month rule, or $3500 by relying on the last month rule. The only reason not to use the last month rule is if you think you won't be eligible next year, either because you change companies or change insurances.
If Turbotax is not offering you the last month rule and is limiting your eligibility to 4 months instead of 7, it sounds like you got the enrollment calendar messed up in the HSA interview. Try running the interview again.
