Even if you are allowed to contribute to your existing HSA, you will receive the best tax benefit by making contributions through a payroll deduction. The IRS says that contributions to an HSA can be from the employer, from the employee through payroll deduction, or from a contribution directly to the account.
Your employer is unlikely to make an HSA contribution if you don't have a high deductible heath plan with them. They also are unlikely to allow you to make a contribution pre-tax from your paycheck, because they don't know that you are eligible o you are not getting the insurance through them.
Making a contribution from your bank account into the HSA would be more costly because the contribution while being tax deductible, will not escape social security and FICA. That means that you will still be "taxed" 7.65% on the contribution.
Having the contribution made from your wife's paycheck will save the most in taxes.
The good news is that you can spend from either account. Some HSA programs allow the funds to be saved in either a savings account, or an investment account. This means if there are no plans to spend those funds it can make sense to try maximize the growth of the funds. Of course investments have risk.