Car Rental in Qatar (Doha)

I have been here in Doha for over a month now. I am a keen traveller and love to explore places. I knew before I came here that public transport is not the best here; but I am still struggling to get off the starting block.

My company are not helping much. I have had conflicting advice but I believe I have the correct story now.

Please correct me if I am wrong; but in order to hire a car here in Qatar I need to have a visa, a letter from my company and a Qatari driving licence. My company allowed my visa to expire and my residents visa has been rejected, which they say is normal.

I know I can drive for a week on my UK licence, but I still don't have my employer's letter.

Can you confrm that I have this process correct and do you know if I can circumvent the system in some way?

Any assistance would be greatly appreciated.