My last reply was probably a bit less than helpful. Let's try again.
For intermediate users not up to making their own stick, but not afraid to edit some files after creation. It is not a one-click user solution for the TinyCorePure64 iso.
One combination that works well on even my crankiest uefi-only hardware is using the front end of this gpl multibooter from 2018:
https://github.com/mbusb/multibootusb/releases/tag/v9.2.0Debs, rpms, source, and even windows exe.
The advantage here is that unlike Ventoy, is that you still have the ability to edit TinyCore's grub.cfg to fine tune your environment when you are done. This can be done easily enough when TC doesn't boot up to the full gui environment, so you use the cli to do so.
The grub.cfg you want to edit will be in the tinycore directory as usual.
If it appears to have non-unix line endings, then you can simply clean them up before editing with
dos2unix [file]
In this case, it would be grub.cfg
In this multiboot environment, it won't be able to find the cde extension directory of course. After initial boot and immediate drop to the cli, you'll want to recursively copy the cde directory to the root directory of your desired target partition. AND don't forget to change the name from cde to tce.
Ideally, make sure your tinycore grub.cfg now points to the tce directory, although it should be picked up automatically if you don't. Study the faqs and posts here for how best to edit your grub.cfg, but this should get one past the initial hump.
Note that this combination works as of 2021 with the 64 bit iso of V12. As with any 3rd party utility, there is no guarantee that this will work with future releases with either multibootusb or TC itself.