If your old e-mail provider uses MS Exchange (like your work may do) you can send a one-time out-of-office message to everyone who sends a mail to you that contains your new e-mail, and you might be able to set an automatic forward to your new e-mail address. If you turn that out-of-office message off and on again, I expect it will reinitialise and send a one-time message again.

Ask your current e-mail provider if they offer such a service.

But there is more to it than just sending such a message. It reaches your contacts, but it won't be processed by a lot of business adresses where they use a no-reply e-mail address. And it certainly won't change your e-mailaddress in all those systems where you use it to login into your account (like CNET). In most cases their site has an option to change your e-mailaddress in your account data after logging in into your account. But even if you don't use your e-mail address to logon, but some user name, you'll have to change your contact e-mail address in your account data yourself.

Your friends, family and relations aren't the problem. All those businesses, clubs and organisations are.