Many rentals show as 'available now' but others in the next month or two. I have seen some lately which a re available in November but no further on.
People don't tend to out their properties onto Rightmove etc too far ahead as they drop down the listings and don't get noticed.
If you contact local letting agents direct, they might be able to tell you about properties which will become available later, but which aren't advertised yet. You could ask if it will be possible to secure them now by paying a deposit. This is more likely to be possible if a property is vacant and having work done to it than if it is currently tenanted.....even if tenants are due to move out at a certain point, because they do t always go when they are meant to, agents usually won't allow a new tenancy to be signed until it is vacant, to ensure they don't have a situation where they have a contract with a new tenant and no house to let them move into.
If you can't afford several months of double rent, you are probably going to have to face some uncertainty. You can limit this a little by contacting local letting agents now and setting yourself up well for the point where you will be ready to rent a new property. Tell them exactly when you want to start renting and that you would be ready to sign immediately and give a very clear instruction of where the property needs to be and ask them to let you know if things become available and ask how far in advance you would be able to sign up and pay a deposit. Then email then and ring on a monthly basis to remind them that you are around and once you get within a month of needing to sign a contract, make contact weekly. This way, you will make the agent aware of your presence and real desire to rent and hopefully be at the front of the queue.
I think you will need to reveal that you are currently in rented and that there might be an overlap. Bear in mind that when credit checks are carried out, this will impact your affordability.....is it likely to mean you fail? It seems quite likely if on an ongoing basis you can't afford 2 lots of rent, as the credit check systems might not be sophisticated enough to factor in that you'll only pay 2 lots of rent for 2 or 3 months.
If there are lots of people approaching letting agents to rent in these popular areas, using the approach I mention above, you won't be top of the priority order.....people who aren't committed to a rental agreement beyond when they want to rent somewhere else are in a stronger position. People who perhaps own a house and have pots of savings are likely to be in a stronger position. You might well be competing against them.
You sound like you'd really be pushing yourself to do double rental.....if your savings are limited and will quickly be reduced to nothing by a couple of months of double rent, then as your DH says, it's probably not a good use of the savings for your family, plus credit checked will be looking at affordability and it will become clear it's not really affordable.
Your timescales and financial position mean you probably won't be able to do this unless your LL and letting agent are willing to release you early....and given they are unlikely to consider doing this until significantly after any point you have vacated the premises and told them in writing you have done this and won't return, and presumably then you won't have anywhere to go because you won't be able to get/afford somewhere else, doing this is not really possible. Unless you have a family memeber willing to sub you several thousand pounds to allow you to pay off your existing contract and vacate, it doesn't sound like this will be possible. In any scenario, costs of several thousand are likely to make it happen becaue of your contracts. And bear in mind that things move slowly, so even if you move out (not sure where to) in order to show LL and agent the property really can be re-let early, the length of time before this appears in paperwork and would show up in any new credit check/affordability check, could be weeks or months...stymying your attempts to rent elsewhere, when time is of the essence.
This renting somewhere else for school admissions works for those with very deep pockets or those who plan ahead with their rentals well well in advance. Unfortunately, you do t have either of these things in your favour.