In addition to the above answer, make sure in your cover letter you are being job specific, and use hints of your own personality to stand out, ( only use professional qualities that you can apply to the position).

Be sure to include ALL of your responsibilities and duties in your previous position as CSS on your resume. They may be looking for a person with experience in one specific task. You could be the only applicant with that experience. Toot your own horn.

You answered your own question

I use my past experience as a Customer Service Supervisor to demonstrate that I already understand the key responsibilities of the role—leading a team, resolving escalated customer issues, training new staff, and ensuring high customer satisfaction. I can give real examples of how I improved service processes, boosted team morale, or hit performance goals in my previous position. Because I’ve already done the job successfully, I bring confidence, leadership, and the ability to handle challenges efficiently. I also understand what it takes to support both customers and team members, which helps me stand out as a ready-to-lead candidate from day one.

Highlight your accomplishments in managing and motivating teams, improving customer satisfaction, and implementing successful customer service strategies in your resume and during interviews. Utilize job search platforms like Jobcase, Indeed, and LinkedIn to find relevant opportunities and tailor your applications to showcase how your past experiences make you a great fit for the role of Customer Service Supervisor. Good luck!