Openrent has been suggested, and it has the advantage of normally dealing direct with the landlord, so you are in a better position to be able to show that you are good tenants.
The last time I had a property to let there were 9 viewings ( where previously if I arranged 10 viewings I would be lucky if 3-4 turned up), everyone asked to take the house. Very few were unsuitable and I ended up letting it to the ‘nicest’ couple as I thought we could have a good relationship. Nearly 2 years later I still thank my lucky stars as they couldn’t be more perfect.
There are some advertisements on Gumtree as well. Just please be careful with Gumtree and Facebook as there are vile scum who try to use desperation as a tool to scam others.
I agree with getting your landlord to write a reference, which you could attach to applications or messages.
Your landlord also needs to be flexible, and if you find somewhere that needs you to take the tenancy virtually immediately, release you from any notice periods, etc.
Your landlord should be aware of how long the court process takes, so is likely to extend the notice period if you are actively looking for somewhere else. Keep in contact with the landlord. Let him know that you are trying.
You will have people saying don’t move until the bailiffs arrive, etc. This really only applies if you have a chance of being housed by the local authority.
Court proceedings are stressful and expensive. If it is hard to find alternative privately rented accommodation now it will be much harder after a court case for eviction and possibly a CCJ for costs.
From what you have said about search area and what you are already doing, I am sure you will find somewhere. If you are currently looking for a 4 person joint tenancy it will put off some landlords, as they may think it is akin to ‘adult sharers’. It might be a good idea to ask for the tenancy just in the names of the 2 employed family members, and have the others as permitted occupiers, or similar.