Without knowing how your customer finds you or the intake procedure, it's tough to come up with the perfect solution for your transition. But I would guess it would start at the sales counter with both existing and new customers by telling them of the services you offer beyond the wraps.
To promote that, I'd suggest you put together a program of the services you want to do, and think of rough timeline of the transition time to do it.
Initially, I don't think a name change is necessary... You can "wrap ink" on just about anything, right? But I would start out with a list of options that you can add on to the wrap design. As you said, things like the websites, printing, and branding. That would be part of your new "Expand Your Brand!" program. Hand the customer a tri-fold with the different ways you can expand their brand. Discuss the options at the sales counter, or let them read through the brochure at their leisure. Follow that up with the proof for the wrap, and the question, "what else do you want?"
Put an "Expand Your Brand!" bug on your website that leads to a gallery showing samples of how the wrap design was used in collateral material.
Devote 25% of your building/street signs to the "Expand Your Brand!" bug --- and the benefit is, it still kinda works as a tagline for your wrap business.
Devote a portion of your interior space to samples of the "Expand Your Brand!" program. Ideally, there'd be an artsy-farsty conference room to host the customers ---- errrrr, clients --- with those samples.
I'd also start using the terminology, "Design Consultation" on your collateral material as a menu or program description. That will help explain to your audience you do more that push a squeegee. That may also work as a sub-head on the "Expand Your Brand!" bug to soften the hesitation of those new to the idea of branding to stop by for an idea of how it works.
As you monitor your success and adjust your business model, you can start to devote more sign space and advertising to the design aspect of the business. Similarly, start moving "Design and Branding" up the list of menu items you advertise, and push wraps and signs lower. If, at some point you want a new name, introduce that as the top line of the company name like, Purty Goode Designs Shoppe home of Wraps Ink.
just my .o2
Admittedly, it sounds as though you are on your way if customers are already asking you for those services. It may be just simply following through on the fundamentals of sales and supporting that transition with subtle changes to your collateral material.