Establish a working relationship with that person. Make a telephone call on any important issue, and do not give up after one call. Two, three or more follow-up calls to track what's happening on a given issue really will get your legislator's attention, since it is rare that anyone takes the time and trouble to follow up. A follow-up letter is also useful. Although most indicated that they often don't know if a constituent who is writing, calling or visiting is a contributor, the staff knows, and your access is through the staff.
Campaign contributions are essential for election and are clearly appreciated.
One thing I noted during my interviews is that physicians have an advantage because of the respect accorded them by society. Thus, they have some extra clout with their elected legislator. You will only increase your access and influence by making a contribution. For the greatest impact, members need to get to know their newly elected legislators early on. The same advice applies to new initiatives or bills in Congress.