Well generally I would just do when I want to do something like upgrade a harddrive without having to reinstall stuff.
I have a couple systems. I'll use netcat and tar to copy over the entire / directory and whatever else over the network to another computer while booted up in knoppix or whatnot. Then once I know that worked out fine, then I'll do the format/hardware change or whatever. Then boot back up into knoppix and prepare the partitiosn or lvm or whatever I am using, copy the information back over.
Then I would need to edit /etc/fstab to reflect the changes, mount the proc directory to the now-restored root partition and install the bootloader. Then reboot and hope everything works. Even if I totally mess up I'd still have all the information aviable stored on another system.
Of course this won't be to helpfull if you only have one system or whatnot.
With LVM it may be even easier if you have enough disk space. Make a new logical volume, bootup in knoppix or whatnot, cp --archive or use tar (recommended) to copy over the OS to the new lvm. Confirm everything is copied correctly then remove the old root volume, then either expand the new logical volume to whatever size you like or recreate the old one with different options. If you changed the names of the logical volumes you'd have to fix it in your bootloader config (I generally have a seperate /boot partition from lvm so that I can use grub easily) and in /etc/fstab, but I think that's about it.
If your using a Redhat system you may have EVM setup. They would be more likely to have advanced features, like maybe a linear/striped conversion utility. I am not sure.. Like I said I am no lvm expert.
I'm a Debian user so I don't know exactly what newer versions of Fedora or Redhat may have.
But if you do find a utility or command option to do the conversion, you'll definately want to backup anything important because it's probably going to be pretty risky irregardless (like if you have a crash or a power outage during a conversion)
edit:
oh and for tar be sure to check out the -p and -s switchs for preserving permissions and order.
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