Should I apply a second coat of primer when I have a high contrast in colors and the new paint is thin?
The best thing to do is have the primer tinted close to the paint color you are using.  For a bright orange paint, the best they can do is probably a pastel sherbet orange color for the primer, but that will still be better than just plain white.

Primer tends to cover well, because thats what many people use it for, and it also tends to be a little cheaper than paint.  Use as many coats of primer as needed to completely cover the base coat before starting with the final color paint.

Another point - some colors cover better than others because of the pigment used.  Colors in the red family (including orange) tend to have very bad coverage. Blues and browns cover much better, so you may not have as much trouble with other colors.  In your case I would strongly recommend taking the primer back and having it tinted.  Many places will do this for free - for instance, at Home Depot they don't even have a way to charge you for it if they wanted to.