Is this for a group of riders or just one rider?
I'll share the experience I had from a big group cross-roads biking event I was part of as a Cyclist, so you can take ideas and maybe create a better plan for yourself.
We were going to cycle across country from north coast to south coast. So first we departed from the country's capital city towards the ride starting point. Two busses for the riders and support personnel, two trucks for the bikes and other items (advertising banners, podium structure, etc). Also a couple of small vehicles.
Once we got to the starting point, the busses served as logistics vehicles, carrying everyone's luggage, and support personnel with food, drinks and tools. The busses departed before us and parked a few kilometres ahead in a previously selected point. (Each bus in a different spot). After a while the busses departed again, overtaking the cycling group and repeating this a few more times. The stops where somewhere like 15-20km (10-15 miles) apart.
The smaller vehicles were also carrying food and tools, but they where going back and forth, sweeping the whole span of the cycling group. And a police officer was riding a motorcycle behind the last rider. The smaller vehicles would drive forward up to a certain point, then go back until they find the motorcycle behind the last rider and then drive forward again, sweeping for stranded cyclists who could not get to the next support point (bus) by themselves. If necessary the vehicle would pick up the rider and bike and take them to the bus. If possible, mechanical or medical support would be delivered by this smaller vehicles.
The whole ride spanned seven days over nine cities. Every stage was from one city to another, (except one stage in which we visited one city just for a few hours ant then continued to a second city). Each day every rider would unload his/her luggage from the busses for the sleep over and load it again the next day.
All the vehicles would be driven at normal road speed and stopped either on planned spots or to help a stopped cyclist, but wouldn't be driven at cyclist speed. The only exception would be the cop's motorcycle at the back of the group.
Each cyclist would ride a his/her own pace, except the pros, who would form peloton. The exception would be that the arrival departure points where a few kilometres outside the city. To go from/to these points to/from downtown, the whole group would form a tight peloton lead by a police car and would travel at around 20km/hr taking the hole lane.
I think this form of logistics worked very well for a group of near 100 riders.
If I was to plan a vehicle supported long ride for a single rider (or a very small group), I would set the car to depart some minutes behind me, calculating to meet me on the road near the mid point of the distance towards a next planned stop point, and repeat.
The distance between the stop points would be what the average of the group would ride in one hour. That means that the car would depart so they meet the group in 30 minutes after the riders started, would drive past us and wait nearly 30 minutes for us in the next stop point.
I hope these ideas serve your purpose and that you are abe to adapt some to your needs.
EDIT: As noted in one of the comments, the support car itself may suffer a failure. It may be obvious to say that the support vehicle shall be in the best operating condition, and must pass a mechanical revision prior to the ride, as with any car-travel activity.
However, I think there are at least two possible alternatives:
1) Is to let the riders depart first and the car attempts to get to the next stop almost at the same time as the cyclists, thus remaining behind the riders all the time.
2) Let the car go first, and the riders would catch up at the next stop. If any rider suffers an accident or a failure for which she/he can't get over, then the support car shall be asked to go back to pick them up.
The planner of the ride must adapt the strategy according to the probability of bike/rider/car failure and to the capability of the riders to fix common issues, or known rider health issues. For example, if no rider carries flat fixing tools, then it is better for the car to reamin behind the riders at all times. But if the riders carry tools and food, then for a little more of a thrill the car can be let ahead of the riders.