Hi Dot
I once made the mistake of telling a Jobcentre Plus employee that I was going on a day trip abroad. She very sternly pointed out that I was required to sign off JSA for that one day and then sign on again (and go through the tedious interview procedures which were in place at that time) on the day after I returned. The fact that I was going away for the day on a Sunday (when there was even less chance of me finding any casual work than normal) was completely irrelevant.
So, if you leave the country to go on holiday (or, indeed, for any other reason) your entitlement to JSA automatically ends. (However you can still sign back on, if there's a gap before your new employment starts, as soon as you return. If you do, make sure that you use the words 'rapid reclaim', which will mean your JSA start date will be two days earlier than the normal process, and there will be fewer formalities to go through).
With regard to your original question:
If you've got employment starting on 1st October, you're fully entitled to claim JSA from the end of your current employment up until then (except when you're abroad). Jobcentre Plus are fully aware that I will start a new job (even though it's on a 'semi-casual' basis) on 7th September. I'm still receiving JSA until then, on the basis that I'll accept any casual work which comes up in the meantime.
Chris