Hi Amy Vann,
in order to understand if it could make sense to add your volunteer experience in the cover letter, let’s start with what a good cover letter should include. Generally speaking, a cover letter should address the following: (i) quick intro on why you are applying; (ii) why you are adapt for the job; (iii) why you are interested in that particular firm compared to others in the same sector; (iv) final remarks, mentioning again your interest.
Your volunteer experience could therefore potential be included in the second area. What you should include there is what can help the recruiter to understand you have the skills to do a great job as a consultant. The kind of specific skills your volunteer experience can show would of course depend on what exactly you did during it, but in general terms it probably helped you to develop better communication skills, proactivity and ability to listen to the needs of people different from you.
The questions you asked thus translate in “Is it necessary that I include a story showing I have good skills in communication/proactivity/listening”? As Vishwa mentioned, consulting companies are interested to see whether people have competences at 360°. The big bias you would have as a PhD would be they would consider you very strong analytical, but not as strong on communication. Thus, a communication/listening story in the cover letter could definitely help you.
As a consequence, my recommendation would be that if you have already better stories showing communication skills, proactivity and ability to listen, then you may exclude the volunteer experience form the cover letter. If not, I would definitely consider to include it, in order to show you have the skills they may expect PhDs lack, as a further point of strength in your profile.