How to best start offering consulting services as a Ph.D. student?
I experienced working as a consultant during a year in a PhD program. 

On the upside, I earned more than my cohort was in our fully funded program. 

On the downside, stepping out of a lecture to take a client's phone call was one of my experiences. Having to handle the fallout of another client's death during finals was another, when one client suddenly became several smaller ones. 

I found I was losing the support of the head of the department, and I didn't like my grades.

I had confounding issues: one was that my clients and wife were in a town a 3 hour drive away from my program and another was that I was scrambling to catch up with the math required, as I underestimated the preparation required. I found the workload to be unsustainable and withdrew, enrolling in a local Masters program instead. 

I'm now likely earning more than the majority of my former cohort, while not facing the stress of the publish or perish monster or dealing with undergraduates trying to understand why you gave them an F for lazy slapdash work. However, I would consider mine a cautionary tale. 

I assume you want to complete your degree. To do so, you'll want the support of your department. You may underestimate their desire to see you become a hugely successful academic eventually rising to tenure at an Ivy. They may become dismayed by your desire to work in consulting. 

On the other hand, there are hugely successful academics who are also very entrepreneurial consultants. 

That said, you don't want your advisor or department chair to think of you as a consultant, unless that's already expected of you. Ideally, you want them to think of you first and foremost as a successful academic. I would avoid loudly publicizing your services until you finish your dissertation (which you should probably finish sooner rather than better, in alignment with this strategy). 

Instead, you might profile your ideal client-type and ask your current clients that best fit that profile for introductions to similar ones, or at least to identify them for you. Then, like any sales pitch, you should do your homework on the potential clients, and go put yourself in front of them. 

I wouldn't rule out the assistantship either. It likely means high quality time with a great scholar. That should align with your values, considering your current efforts at education. Don't be dishonest about your intentions, but do be discrete. I hope my advice helps.