Install wget, if you don't have it already installed.
Download the iso image. For this example, let's say you download it to /home/tc/Downloads. Let's say it is tinycore_3.4.iso. If doing this in the future, change this to the latest version.
In the terminal type:
cd /home/tc/Downloads
wget -c http://distro.ibiblio.org/pub/linux/distributions/tinycorelinux/3.x/release/tinycore_3.4.iso
After downloading is finished:
Make a new directory in /home/tc/Downloads. For this example, let's call it tmp.
Open the terminal and type:
sudo su
cd /home/tc/Downloads
mount -o loop tinycore_3.4.iso tmp
You can now copy Tiny Core from the directory tmp.
For this example, let's say you have Tiny Core on partition sda1. If you have it in another partition, change this to the correct partition.
Create a new directory in the partition /mnt/sda1. For this example, let's call it tinycore_3.4.
Using a root file manager, go back to /home/tc/Downloads/tmp/boot. Copy bzImage and tinycore.gz. Paste them to /mnt/sda1/tinycore_3.4.
Edit your bootloader. If you are using Grub legacy, it may be something like this:
title Tiny Core 3.4
root (hd0,0)
kernel /tinycore_3.4/bzImage quiet opt=sda1 tce=sda1 home=sda1
initrd /tinycore_3.4/tinycore.gz
title Tiny Core 3.3
root (hd0,0)
kernel /boot/bzImage quiet opt=sda1 tce=sda1 home=sda1
initrd /boot/tinycore.gz
If you are using a different bootloader, edit as appropriate.
Turn the computer off, and restart. You can now run either Tiny Core 3.3 or 3.4.
With updates with some versions, there are other files which also need to be changed. See the release notes for each version. With this version, it should work without changing other files.