Absoleutely, onwards...........
Council aside. I'm sure (if HB finances allow), that you will get another privately rented house.
You seem as if you're educated and self motivated, so many people would be moaning its not fair and expecting help from everyone... You need to give yourself an advantage and stand out from other tenants, be that working tenants or not. You do have huge advantages, your renting history, your stability, your age (sorry, making assumptions based on 3 kids).
I said ringing Agents before, but perhaps it would be better to spend an afternoon going in to see them.
Put together a 'professional' pack to approach Letting Agents with. Such as: your ex Landlord's reference
the current one's reference and the reason you are having to leave
details of what HB you are entitled to and how much top-up you will be paying, 3 months of bank statements,
3 months pay-slips,
maybe an employers reference (if working).
a covering letter which sells you and gives your contact details - take several copies with you and leave this with the Agents.
If you have any CCJs or anything that would mean your credit referencing wouldnt be good, be honest about it. Can you get a rental guarantor, someone who works and owns their own home and agrees to pay rent should you not be able to?
The more you can impress the letting Agents, the more they will sell you to Landlords and believe me what the Agents say matters (you were polite, well spoken, clean, tidy...) probably more than credit referencing. This may all seem obvious, or OTT, but as a Landlord, you dont get tenants like this very often 