Yes, you should do if you can do it without exceeding your annual contribution limit. By taking money from your checking account and putting it in the HSA you are making a contribution to the HSA. You will be able to get a deduction for the contribution on your 2024 tax return. You can then take money from the HSA to pay the medical bill, so you will be paying the bill with tax-free money.
BUT - It doesn't matter that your HSA contributions, or the balance in your HSA, are below the annual limit now. You have to make sure that your total contributions during the year do not exceed the limit. Your total contributions include payroll deductions, employer contributions, and any additional contribution that you make from your checking account. Taking money out of the HSA to pay a medical bill does not mean that you can contribute more. What matters is the total amount that you put into the HSA during the year, regardless of how much you take out, not the balance in the HSA at any particular time.
If you do not have enough money in the HSA now to pay the medical bill, you can pay it from your checking account. Then later on, when there's more money in the HSA, you can reimburse yourself for what you paid by taking money out of the HSA. You still get the same tax benefit.
