@ES808 Thanks for reaching out!
Based on the measurements you have given us, it sounds like your son is right there at the top of the scale for a 20" bike and just at the beginning stages of fitting a 24" bike. The 24" bikes we carry by Co-op Cycles, Salsa, and Cannondale fit a height range of kids from 46" to around 60". So, in terms of seat height, you should be good to go there. The real concern is going to be the standover height. Here is the rundown of 24" bikes we carry and their standover height:
- Co-op Cycles REV CTY Step-Through Kid's Bike. Standover Height: 22.8"
- Co-op Cycles REV 24 Plus Kid's Bike. Standover Height: 22.25"
- Cannondale Cujo 24 Plus Kid's Bike. Standover Height: 23.9"
- Cannondale Quick 24 Kid's Bike. Standover Height: 23.1"
- Salsa Timberjack 24 Plus Kid's Bike. Standover Height: 23.3"
This is going to mean that your son will be at the very minimum of clearance over the bar, which will likely result in some awkward dismounts and potentially some challenges getting started at first, depending on his experience on a bike. If he is an active rider without training wheels and needs simply to upgrade his size, then you should be totally fine with one of these 24" bikes.
If he is a beginning rider, or not quite fully confident in his riding abilities, then you may want to consider starting with a smaller bike at first and working up to a bike this big as he gains confidence and ability. As an example, the Salsa Timberjack 20 SUS Plus Kid's Bike fits a height range of 49" - 57" with a standover height of 18.4". Likewise, the Co-op Cycles REV 20 6 Speed Kid's Bike fits the same height range and has a standover height of 20.5". Ultimately, you know your son and his abilities best. At this time, it appears as though all of these 24" bikes are currently in stock.
Hopefully this helps, thanks!