Your e-mail is good, although Linkedin will have the low conversion rates. Here are some other ways to get a referral:
Here are some tips on how you can do that:
1) Leverage your own network - find friends or friends of friends who can practice cases with you or make you a reference.
2) Attend company events. Consulting companies do a lot of events both for graduates and experienced hires. Find the social network groups related to consulting or websites, subscribe to newsletters and stay tuned. Also, check if they have events in your University since you are a PHD.
While some of the events will be open to everyone, others will require a resume and a cover letter, so make sure to prep.
3) Talk to people on the events and send Thank You notes. After each event, there is a Q&A session where you can talk to consultants 1 on 1 or in a group. Ask for contact info or send a thank you note after the event ("I just wanted to thank you for visiting our University... It was especially interesting to hear about... Would be happy to keep in touch and apply in the nearest future.). Alternatively, you may use linkedin for that. If you are an experienced hire I strongly suggest to ask for a 1 on 1 meeting in a thank you note.
4) 1 on 1 meetings via Linkedin. All people like giving an advice. So don't hesitate to ask consultants for a career advice. Tell your story and ask how consulting fits into it.
5) Mock interview Depending on where you are in your prep process you may ask a consultant for a mock interview. Consultants are very much opened to help even if it is a cold call e-mail. The main problem is a lack of time on their side. So don't be afraid to remind about yourself if the consultant has already committed but finds it hard to find the time
6) Talking to partners If you are an experienced hire I suggest to talk directly to the partners in your industry or the partner responsible for experienced hires (in consulting partners also have additional roles). Get an intro from the HR or from your friends working there. Partners care a lot about experienced hires with relevant industry expertise. Chances to get referral are much lower though. They also will not be interested in giving mock interviews.
7) If you are an MBA graduate Just talk to your section mates and ask for a reference or a mock interview - they will be happy to help. Make sure you attend all the consulting events.