Unetbootin uses syslinux to create a bootable linux filesystem. After you ran it for your usb key (assuming it is FAT on your key), browse it and create a folder structure tce/optional/. Put into optional all your desired extensions. Create a file called onboot.lst in tce/, which contains your extensions to load on startup.
Example (this is linux like, if you have windows, then create dirs and files within windows explorer, download the extensions in a browser, etc.):
cd /media/USB
mkdir -p tce/optional
cd tce
echo kmaps.tcz > onboot.lst
cd optional
wget http://distro.ibiblio.org/pub/linux/distributions/tinycorelinux/3.x/tcz/kmaps.tcz
wget http://distro.ibiblio.org/pub/linux/distributions/tinycorelinux/3.x/tcz/kmaps.tcz.md5.txt
Touch your backup file:
cd /media/USB
touch mydata.tgz
Open the file /media/USB/syslinux.cfg. Edit the tinycore entry. Example:
label ubnentry0
menu label tinycore
kernel /boot/bzImage
append initrd=/boot/tinycore.gz quiet
After or before quiet you can add boot codes. Example for a changed line:
append initrd=/boot/tinycore.gz quiet waitusb=6 tce=sda1 restore=sda1
Waitusb is necessary because usb devices need some time to be initialized on startup. The entries are assuming (sda1) that your usb key will be recognized by tinycore as the first scsi/usb/sata disk in the system. If you have other drives you may need to change it to /dev/sdb1 or /dev/sdc1, ...
Save your syslinux.cfg and plug the key in, boot from it. Take a look after a boot if the kmaps extension was loaded (must be present in /tmp/tcloop).