This is an often-overlooked question.
The first step in rolling out a new brand identity and avoiding alienation of your existing customers is to tell your current and past customers about the change. After all, how would you feel if you went to the website of a company that you'd worked with, or purchased something from, and it looked completely different than you remembered?
Whenever this happens to me, I wonder if I'm in the right place – and I usually wind up looking around the site for familiar-sounding bits of text, or going back to Google and searching for the right website. Which winds up taking more time and being a frustrating experience – especially if I was on the right site in the first place.
When you change your brand identity, make it a point to announce the change to your past customers. Send them an email or a newsletter at the very least. Mailing a postcard or making a nice announcement can make it seem like a more formal launch. You could even have a party. Just do something to make your customers aware, and make sure that whatever you do is a good match for your brand's personality (i.e., don't be too formal or playful about it, depending on what the feel of your new brand is like).
If you really want to make your best customers feel like a part of the change, then why not make them a part of the design of the new brand?
A small business's brand will be more effective if you design it with your target audiences' – and best customers' – design preferences in mind. You want to create designs that resonate with them, that speak to their souls, and that tell your business's story to them at a glance – in their visual language.
During the design process, it can really help to have a few of your best customers on-board, helping out by providing input and feedback into the process. Send them your new brand summary, so they know what they're evaluating the designs for (in other words, don't just send them a bunch of logos and ask them which one they like – because appealing to their personal preference is very different than creating designs with meaning). Then, send them the various logos, and ask for their input. They'll usually feel honored to be a part of such a big decision, and be happy to provide feedback.