If you are covered by the HDHP on 12/1/24 AND plan to remain covered by a qualifying HDHP for all of 2025, you can use the "last month rule" to contribute up to the maximum amount of $4150. Your employer is setting your limits based on the known facts of your employment ($345 per month) without assuming that you will remain covered or employed (you might change coverage, or be let go, or you might get married or have a child which would increase your limit).
If you don't rely on the last month rule, your limit is $345.83 for each month that you are covered by a qualifying HDHP on the first day of the month. Assuming your insurance started 3/1/24, your limit would be $3458. However, if your insurance started on March 15 (for example), then your limit for 2024 (without using the last month rule) would be $345.83 x 9 months = $3112.
You don't need to change anything now. If you get to December 2, and you are still covered by the HDHP and you think you will be covered for the whole next year, you can make an extra contribution out-of-pocket, from your bank account to the HSA, to "top it off" to the full $4150, and get an extra tax deduction for the contribution. If your HDHP coverage ends sometime before the end of the year, you have until April 15, 2025, to recalculate your actual contribution limit and remove any excess contributions. (This is a special procedure at the HSA bank, not a normal withdrawal.) If you remove the excess contributions before the deadline, they will be added back to your taxable income (since they are not eligible to be tax-free) but there is no additional penalty.
