After running 'make modules' and 'make modules_install', if everything went well, the new modules will be found in /lib/modules/. It is always a good thing to rename the old module directory to something else, e.g. "modules.OLD", before running the above commands, otherwise the old directory will be overwritten.
Once you have the new modules, you need to unpack the tinycore iso, unpack the tinycore filesystem (tinycore.gz) found in the iso, replace the old modules with the new ones, then repack the filesystem and finally create a new iso. I gather from your previous post that you already know how to extract the files from an .iso and repack it. I'll post the whole procedure here, just in case.
To get the tc filesystem (tinycore.gz) and all the other files (which you will need later) from the iso:
mkdir /mnt/tc-iso
mkdir /tmp/tc-filesystem
mount tinycore.iso /mnt/tc-iso -o loop,ro
cp -a /mnt/tc-iso/boot /tmp
mv /tmp/boot/tinycore.gz /tmp
umount /mnt/tc-iso
Now extract the tinycore.gz filesystem:
cd /tmp/tc-filesystem
zcat /tmp/tinycore.gz | sudo cpio -i -H newc -d
Now, remove the old modules in /tmp/tc-filesystem/lib/modules and copy there the new ones.
[By the way, and this is optional, if you wish to compress the new modules from x.ko to x.ko.gz (which is tinycore standard) you will need to use the program advdef, which is part of the advcomp.tcz extension. First gzip the new modules with 'gzip -2', then advdef them with 'advdef -z4']
Having replaced the modules, you are ready to repack the filesystem, thusly:
cd /tmp/tc-filesystem
find | cpio -o -H newc | gzip -2 > ../tinycore.gz
cd /tmp
advdef -z4 tinycore.gz
Now move the tinycore.gz compressed filesystem to the boot directory and create a new iso:
cd /tmp
mv tinycore.gz boot
mkdir newiso
mv boot newiso
mkisofs -l -J -R -V TC-custom -no-emul-boot -boot-load-size 4 -boot-info-table
-b boot/isolinux/isolinux.bin -c boot/isolinux/boot.cat -o TC-remastered.iso newiso
rm -rf newiso
Note: the mkisofs command line is shown above in two lines (the 5th and 6th lines), but you need to enter it as ONE line.
This covers the whole process needed to unpack the iso, unpack the compressed tinycore filesystem, replace the modules in the tinycore filesystem, then recompress the filesystem and create a new iso.
Hope it helps.