Yes, you do not need to reinstall. I have never done this, but I think what you need to do is
Mount your new disk as /mnt/new
as root, and probably in single-user mode, you need to copy the whole contents of /home to /mnt/new, preserving ownership and timestamps of the files. I think the command would be
cp -pr /home /mnt/new
Then, in single-user mode, you need to rename /home as /home.old (in case you/I have made a mistake).
Then unmount the new disk from /mnt and then mount it as /home.
Restart multi-user mode. If everything is OK, you need to edit fstab to mount the new disk as /home at boot time.
Someone will probably make corrections for me here!