Leaving aside selling a car into the trade, paying twice for one month of tax cannot be avoided unless the vehicle is declared SORN.
Using April/May as an example, If the Registered Keeper (RK) is changed on the last day of April, the old keeper (O) will not get a refund for that month. The new keeper (N) - having acquired it during April - will have to pay tax from the first of April.
The way to avoid this is for O to declare the car SORN on the last day of April (or earlier if neither he nor N want to use it), ensuring it is kept off road. O will not get a rebate for April. However, provided the SORN is not removed until the first day of May, N will only have to pay tax from the 1st of that month.
I've been through all this before. As far as I can see, the only way to avoid losing a month's tax one way or the other without declaring it SORN would be to change the Registered Keeper precisely at midnight on the last/first day of a month. However, “midnight” does not exist for these purposes. It is a fleeting moment in time of no duration which divides two days. When the change is made the DVLA’s clock will either show 23:59 on the 30th April or 00:00 on the 1st May.
Of course the new owner could simply try taxing the car from the 1st May and he may get away with it. But he shouldn’t.