Places to consider
Hi Boriss,
I know nothing about Lee-0n-Solent.
However I can vouch for both Thruxton and Old Sarum, both of which should be reasonably convenient to where you are located with good instruction standards and reasonable prices.
Goodwood is very expensive.
Booker is a very long way from you: I would suggest that you consider White Waltham rather than Booker although White Waltham is still quite a long way away from you and very expensive.
Fairoaks too is a very long way from you and I have never heard anyone praise the clubs based there. (I had a bad experience at one of them, but that was several years ago; things may have changed).
Blackbushe might be worth a look if you use Redair. Redair also operate from Redhill, where they have always had a good name over very many years. There is, however, another outfit at Blackbushe about which I know nothing.
Personally, I would not use any establishment that takes me more than an hour's travel to get to. You can all too often drive there and find the place clamped with bad weather and then you have to drive all the way home having achieved nothing but an expensive petrol bill.
'Hope this helps.
BP.