Unfortunately, you are considered "covered" by a workplace plan for the entire year even if you were not employed by that employer for the whole year. As such, you follow these rules.
You can always contribute to a traditional IRA up to $6500 for 2023 (or $7500 if over age 50). However, you can only take a tax deduction if your income is under the limits on this chart.
https://www.irs.gov/retirement-plans/2023-ira-deduction-limits-effect-of-modified-agi-on-deduction-i...
If you make a contribution that is not deductible, that creates a situation that you will have to keep track of for the rest of your life, to avoid paying tax on those contributions when you withdraw them.
Your ability to make contributions to a Roth IRA are covered by this chart.
https://www.irs.gov/retirement-plans/amount-of-roth-ira-contributions-that-you-can-make-for-2023
A Roth IRA is also not deductible when you contribute, and you don't pay taxes when you withdraw, and there is less paperwork hassle than making non-deductible contributions to a traditional IRA.
