**You can apply again through Greece if Greece is your genuine main destination.** A previous refusal does not automatically require another refusal; your new application must be assessed on its merits. [EU Visa Code, Article 21(9)](https://eur-lex.europa.eu/legal-content/en/TXT/?uri=CELEX%3A02009R0810-20240611)

For a tourist trip, apply through:

- **Greece** if it is your only destination or where you will spend the most time.
- **Another country** if that country is genuinely your main destination.
- The **first Schengen country you enter** if your stays in the different countries are equal in length. [European Commission guidance](https://home-affairs.ec.europa.eu/policies/schengen/visa-policy/applying-schengen-visa_en)

The priority is addressing the refusal reasons. They indicate that the consulate was unconvinced by your explanation or evidence of the trip. For your new application, follow the consulate’s checklist and provide a clear itinerary, verifiable accommodation and transport reservations, evidence of funds, and evidence of your intention to return home. Make sure dates and details agree across all documents. A brief cover letter explaining the previous refusal and how your new evidence addresses it would be sensible. [Document requirements](https://home-affairs.ec.europa.eu/policies/schengen/visa-policy/applying-schengen-visa_en)

Your UK visa does **not guarantee Schengen approval**. If you complete the UK trip before applying, you can include evidence of that travel and timely return as additional travel history.

My advice is to choose your actual destination first, then strengthen the application. Switching countries alone will not resolve the concerns raised in your refusal.