Directly setting number of supports to a number higher than 7 is not going to work without modification to the structure of support data, which is always 24 bytes in length regardless of the number of supports; 7 each for partners/initial value/growth rate and 1 for number of support partners, then 2 bytes of padding to keep things 4-aligned. Setting # of support partners to 8 will just have it read from (First character ID + 8) for the 8th partner, which is the first byte of the next set of 7 data.
That being said, as long as a character would not be able to possess supports with more than 3 characters at one time then all you would have to do is change support data to a list with a terminator so it can be variable length, then change how the game loads support data into the character struct to load the character ID in along with the support value, which given 7 bytes to work with there is room for in the existing struct. When the game saves this edited support data it would save and load it properly with no modification on that end since it’s still saving and loading the values of all 7 bytes, and everything would work fine.
If you want a max of 8 simultaneous supports, the easiest way would be to 1. Restructure support data to have a max of 8 support slots and 2. Use the byte that’s normally a bitfield for which supports you can’t rank up again on that same map as the 8th support byte, as well as disabling the normal check on that byte. This likely would not require editing save data either, at most it would limit your 8th support partner to a character ID of 0x7F or lower if it doesn’t pack the 8th bit of that byte in vanilla, and even if it doesn’t it’s an incredibly trivial thing to avoid and one you likely would never run into as all playables and bosses generally fit under 0x7F.
If you want more than 8 and/or are intent on keeping that feature, you’ll either need to find free space elsewhere in the character struct (there is little to none) or put it elsewhere in RAM.