So, my question is ... Is the S29GL032M (or AMD equivalent AM29LV320M) already supported in byte mode for any board in u-boot?. I have been testing some of the drivers, but I still couldn't find a working solution for my board.

I think there isn't a flash driver for S29GL032M in BYTE-Mode, so you have to write your own driver. Be careful, because the AMD AM29LV320M supports a BYTE-Programm mode, to write a single byte into flash. This mode is used in many (AMD) flash drivers in U-Boot. The S29GL032M doesn't support this mode, so you have to use write buffer programming if your Flash is in X8 (BYTE) mode.

Best regards Sebastian