I presume that this is actually about primary schools, not secondary.
I understand that you don't want to send your son to the school you have been offered. However, if you reject the school the LA is not under any obligation to come up with another offer. If they do, it is likely to be at another unpopular school even further from home. Your son must be in full time education by the start of term following his fifth birthday so you need to have a plan.
As your son is not yet 5 you can defer entry for a full year. The LA may allow you to defer and apply for Reception next year but it is quite likely that they will insist on him going straight into Y1, missing out on Reception completely. Even if they do allow you to apply for Reception, there is no guarantee that the situation will be any different next year. You may still end up with an offer you don't want.
You can enquire about other schools with places and apply for them if you want. However, if other people are doing the same you may not get a place and the LA may insist that you come off the waiting list for your preferred schools if you want to apply to other schools.
If you move closer to one of the schools it will move you up the waiting list provided it is a genuine permanent move. If you own the property you currently live in, the LA is likely to continue using that as your address even if you rent near your preferred school. Yes, you can have many houses but the LA is entitled to decide which address to use for your application. If you own and rent most will go for the house you own.
You can look at other schools and you can hope to get a place at one of your preferred schools through the waiting list. You can also appeal for your preferred schools, although the chances of success at appeal are generally low as most cases are heard under infant class size rules.
As others have said, you should accept the school that has been offered. Rejecting it will not improve your chances of getting a school you want. It will simply leave you without a school for your son.