Be polite to the person who answers the phone even if you’ve disagreed with their boss in the past. Let them know that you ARE their constituents. Legislators are only responsive to constituents from their own districts.
Thank them for their time and something specific. Starting with thanks will make it easier for them to hear your concern; however, keep it brief as you transition into what you need.
Briefly tell them what is important to you and why. Be specific, compelling, and brief.
Make a specific request and take note of the answer so you can follow up with them.
Thank the person for their time and remind them of the answer they gave you and the specific request you made. The more times you are able to repeat your request, the likelier it is that it will get through to them when the time comes to act.
Follow up. Follow up with a hand-written note, or an email, thanking them again and reminding them of your specific request and the answer they gave you.