I am a management consultant that focuses on what we call "Financial Effectiveness." What we do is largely process-driven and focused on streamlining the finance function for our clients. Our clients include corporate retailers, industrial products and service providers, and financial services institutions. While we work in the finance function, we mainly add value by thinking through the different processes, recognizing inefficiencies, and implementing new technologies or strategies. I think this might sound like what you are looking to do, so please know that the job exists out there! Begin to research large firms (such as "the big 4," being KPMG, PwC, EY, and Deloitte) as well as smaller consulting firms that might specialize in Corporate Finance. This is an area where you can provide management consulting services, while utilizing your finance skills.
That being said, I came into this position without experience in management consulting in particular. Most firms (especially the larger ones) are confident in their internal resources and the ability to train their new hires. What is important is that you make connections with people in the firm you are looking into, and are confident in your interview skills. Specific management consulting experience is not necessarily required for lower level positions.
Hope this helps! Best of luck to you!