I would try SeaMonkey. It is based on the older Netscape browser and may be compatible with Windows ME.
You would not have to do anything with your IE6. Windows will quite happily work with more than one browser installed and they will not interfere with each other.
However, I wonder if you need to do anything. It is true that Internet Explorer is considered the least secure of today's browsers but for an older operating system like ME the threat of viruses, malware, or hacking is diminished. Such malware writers are targeting the newer operating systems now. So as long as you take simple precautions, eg a firewall, an anti-virus, an anti-spyware, and be careful with your personal information, I would hazard a guess that you and your system will not be too bothered by such considerations.
My PC is a dinosaur with Windows ME and Microsoft Internet Explorer 6. I have been advised to download a different browser, or rather update to a different one. I do not know which browser would be compatable with my dinosaur. Also once I find a different browser, do I need to uninstall the old Microsoft IE?
My PC is a Compaq Presario 5000 Series, 20.0GB hard drive, Windows ME, 56K modem,(yes, with times as they are, my budget can only afford dial up) 192.0 MB Ram ( I increased the ram when I installed a burner years ago).