I’m not 100% clear on what you are trying to do here. Are you saying that you want a new email address that would be something like your_name@the_organisation_in_question.com? If so, it would be up to the organisation to set you up with an address on their domain; you can’t set up a Gmail account with a non Gmail domain name as far as I know.
However, what you can do is use the Gmail web client (or mobile apps) to access email from other services as well, as long as they have POP3 access. In fact, you can use the Gmail account you already have and add the new organisation’s email account to your inbox - and therefore manage all your email in one place. You’ll still need the organisation in question to set you up with an email address, though.
If that’s what you’re after, this is how you do it.