Assuming your new iso is in /mnt/sda1/iso and your boot files are in /mnt/sda1/boot:
$ mkdir /tmp/image
$ sudo mount -o loop /mnt/sda1/iso/tinycore_1.0rc9.iso /tmp/image
$ sudo cp /tmp/image/boot/tinycore.gz /mnt/sda1/boot
[check the two sets of file match]
$ ls -l /mnt/sda1/boot
-r--r--r-- 1 root root 2021808 Sep 2 21:26 bzImage
drwxr-xr-x 2 root root 4096 Dec 27 21:36 extlinux
-r--r--r-- 1 root root 8632308 Dec 31 12:08 tinycore.gz
$ ls -l /tmp/image/boot
-r--r--r-- 1 root root 2021808 Jul 24 00:23 bzImage
dr-xr-xr-x 2 root root 2048 Dec 23 02:52 isolinux
-r--r--r-- 1 root root 8632308 Dec 30 21:33 tinycore.gz
[ensure the file is completely written]
$ sync
..and re-boot