You can only contribute to a 403(b) or 401(k) via payroll deduction, you can't make ad-hoc contributions.
However, if you are separated from service, you can roll the account balance over into a personal IRA, or into the retirement account of your next job. Make sure you do a direct rollover, don't get a check. (You can also leave the money in the 403(b) as long as you want, if you are happy with the plan. Company plans usually have limited investment choices but lower fees.)
Whether you leave the money in the 403(b) or roll it over, you can contribute up to $5500 to an IRA ($6500 if you are 50 or older). But, the IRA contribution might not be tax deductible, depending on your overall income for the year and filing status, because you were covered by a retirement plan at work. See this link,
https://www.irs.gov/retirement-plans/2017-ira-deduction-limits-effect-of-modified-agi-on-deduction-i...
If your deductible IRA contributions are limited, I would not make non-deductible contributions to a traditional IRA, that mixes taxable and non-taxable income and will confuse things when you retire. Instead, you can contribute to a Roth IRA. Roth IRA contributions are also maxed out at $5500 or $6500 depending on age, and may be reduced depending on your income and filing status. See link. https://www.irs.gov/retirement-plans/amount-of-roth-ira-contributions-that-you-can-make-for-2017
