Hello xerez,
Sorry for the probably late response to you post, but I recently succeeded in dual booting Slackware and Windows 10.
If you want your BIOS to give you the option to boot the Linux, GRUB must be present on the EFI partition and you should modify the UEFI settings to list the option there.
What you need is to know which is your EFI partition (e.g. /dev/sda2) and efibootmgr (utility to access and modify UEFI settings).
So, insert your live CD and boot into your newly installed Linux Mint (on hard drive). The live CD only serves as bootloader in this case. In the following steps, it is assumed that the EFI partition is /dev/sda2 and it is mounted at /boot/efi. # is the prompt for root.
# mount -t vfat /dev/sda2 /boot/efi (in case the EFI partition was not mounted automatically)
# modprobe dm-mod
# grub-install --target=x86_64-efi --efi-directory=/boot/efi --bootloader-id=grub --debug --recheck
# grub-mkconfig -o /boot/grub/grub.cfg
The above must have installed grub on the EFI partition and generated grub.cfg (where the parameters for the grub menu are defined). Just check if everything is in place on the EFI partition. We will assume the above steps placed a file at /boot/efi/EFI/BOOT/grubx64.efi
Now you need to modify the UEFI settings:
# modprobe efivars
# efibootmgr -c -g -d /dev/sda -p 2 -w -L "Linux" -l '\EFI\BOOT\grubx64.efi'
Notice that the option "-d /dev/sda" refers to the hard drive and "-p 2" refers to the EFI partition.
I hope the above will work for you. There are a couple of things you might need to change to adapt it to Linux Mint, for which the underlying system might be a bit different. If the BIOS does not seem to grab the grubx64.efi, try to change the path to 'EFI\GRUB\bootx64.efi' and repeat the efivars and efibootmgr steps.
All the best
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