Breaking into Healthcare Consulting

Long time lurker but new poster,

I am currently interested in healthcare consulting. I currently am a healthcare provider with about ~1 year experience (not MD or RN). I also have an MHA degree from a non target school. I am currently employed as a provider managing an outpatient clinic as well as performing provider duties. I am possibly looking to leave clinical side d/t the burnout and frustrations that have burdened providers.

I have a strong interest in business.  I have technically done free consulting for private practices on growing their businesses. I developed a report on ways these practices can grow their business's after interviewing many in the community. I eventually presented this report at a national conference as the practice owner's were very impressed. I am not sure how much this experience would be of any interest to consulting firms. My decision to get a MHA was so I can transition into more managerial/consulting/administrative roles within healthcare.

I was hoping for some advice about transitioning to a consulting firm, what firms would be good for providers and if I should switch to consulting or stay in my current position and grow within the company. My current company is a larger outpatient provider, PE backed. Looking to grow substantially in the next several years. I worry about insurance reimbursements limiting income potential even if I reach managerial levels.

Obviously I would like to stay in healthcare, if I switch to a consulting firm and my MHA limits me to that field, which is fine as this my area of interest. 

Would it make more sense to stay within my current company, climb the ladder to get more experience in the industry then transition to consulting? Or make a switch sooner?