When is the best time for job hunting in the Gulf?

there is no (best) time for job hunt, because its not the time that affects your candidature towards a job but what you carry in your resume.
and as far as willingness to relocate is concerned, it does help your application to be considered on a comparatively much broder spectrum.
wish you luck if you are looking for a job.

As an active job-seeker who is seeking job opportunities in the Gulf, I have a few things in mind,

The best months to look for a job are September and January-February.

In these months, people are returning from vacations and are willing to hire new talent.

The worst time is Ramadan and summers.

If you are flying in from other countries, then it is important that you do your homework 6 months prior from coming in.

Apply for jobs online and understand the market stats.

Stay active on LinkedIn and build a network.

Talk to the people of your field and HRs specifically via LinkedIn messaging on how to initate a job search?

I got a lot help from doing that.

If you are qualified and lucky enough you might get interviews on Skype before flying in.

And your job search will be easier than ever, because you know the market insights even if it is from outside.