If you go abroad, you need to sign off just before you go. When you get back you may be able to make a rapid reclaim for JSA online which should only take 10 minutes.
However, it is likely that when you return from Tenerife and try to make a claim for any Income-based JSA, the Jobcentre Plus will tell you that you have to claim Universal Credit instead.
Perhaps you can ring them up, speak to a Supervisor to see what can be done to help you? Be prepared though to be told that it is your responsibility to keep them informed of changes to your circumstances and that you will have to apply from the beginning, again when you get back. I do hope the last phone call between you and them was amicable?