I don't think any employment adviser, in their right mind, @ 'JCP' is going to go out of their way to prevent you taking up a new post if youv'e got one. (There's that much flack flying around over unemployment stats - especially with 06.05 looming!). - Besides, you'd be providing a heaven-sent opportunity for your local 'JCP' to "crunch" their monthly stats!
Re; staying on where you are - even for a short time - you'd be surprised how many caretaker opportunities exist in such situations. Employers, mainly, just muddle through in the interim - but if you don't ask 'em, they won't tell you, I'm afraid.
Go for it & good luck.