I'd say the majority of sponsored rides are not suitable for beginners, you need to be secure & in control in walk, trot, and canter plus capable of staying in control if the horse gets excited. You might get groups coming past you in canter & that's enough to hot up the quietest of horses .
Even if you feel you're capable finding a horse may be difficult and expensive. The insurance issue may mean a school is reluctant to hire out, and even if they are you're going to have to pay a full day's hire as the horse won't be doing lessons before or after, probably a premium in case there's any injury, possibly for a member of staff to accompany you on another horse, transport to and from the ride & entry fees.
I think you'd probably enjoy yourself far more booking a long hack either at your riding school or somewhere else - it would be safer, cheaper and easier to arrange.