You can make post-tax contributions at any time throughout the year.
At the end of the year, you will receive Tax Form 8889 with the total amount of contributions you can claim on your tax return as tax deductible. One advantage of contributing via payroll (through your employer) is that any contribution made will not be subject to Social Security and FICA taxes – this amounts to a 7.65% savings. If you earn more than $127,200, this benefit won’t apply since that is when Social Security and FICA phase out.
Source: IRC Sec. 223(a).