Can I receive unemployment benefits after just finishing my terminal leave?

Possibly, but ultimately the State's law (the one you'll reside in after terminal leave) determines eligibility requirements and conditions.

The overarching requirements are in the Unemployment Compensation for Ex-Servicemembers (UCX) program*. Generally, if you were honorably discharged from active duty after your first or subsequent term, you'll be eligible from a "federal point of view", but you'll have to meet the eligibility for the state.

You won't know until you submit with your state* for unemployment compensation.
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* UCX Fact Sheet - https://oui.doleta.gov/unemploy/docs/factsheet/UCX_FactSheet.pdf
* UCX detailed handbook - https://www.dol.gov/agencies/eta/advisories/handbooks/et-handbook-no-384
* Unemployment offices - https://www.careeronestop.org/LocalHelp/UnemploymentBenefits/unemployment-benefits.aspx

Your first stop needs to be the Veterans Services Officer in your county of residence. They work for the county and are the best source for Local, State and Federal benefits.

You will have to claim it in the state you are in. I have 2 breaks in service. Each time I went on unemployment. I was living in NC both times. There was a wait period to start getting paid. I went to the office and filed. As soon as my leave was off I was in the pay period for my unemployment. I would look into your state's website for more info.