He won't get into private school - unless you pay like everyone else! There's no freebies for smarter kids.
If he enjoys it, just let him be. He's happy. He doesn't need more and more. No kid ever looked back on their childhood and said "gosh I wish I'd had some harder maths problems." Your house has books, he knows where they are if he wants to move on. The school will challenge him, ensuring he hasn't skipped any vital building blocks (you'd be surprised.) There isn't anything to actually DO about it.
Honestly, 'giftedness' isn't really a gift. If the child is intelligent and enjoy learning they will simply enjoy school and life and live happily ever after, but it's no more amazing than loving and being quite good at tennis or poker.
A child with a too-high IQ, whose brain just rushes around 'learning' and gobbling up information because they find solace in the patterns, who becomes frustrated with other children, people, the whole world, because it doesn't fit into logic and pattern and their models, then it starts to become a learning disability all of its own. Then it becomes a curse.
My son's got an evaluation next week - whether for Asperger's, asynchronous development, 'giftedness' (cringe), we don't know yet. Believe me. All that early reading, precocious verbal skill, deep conversation, complex thought processes, word play and wit... we didn't see it as a joy. We saw alarm bells. We just knew the path we were starting on. The school noticed within weeks. He's just about to turn 5 and even though I don't mention it to him, he's noticing it himself now.
Enjoy it, and just hope he likes numbers and words. And that he evens out, his development stays on track and can enjoy being a child.