How might I create accurate regional maps when starting from a world map?

[ChargingBadger]

Here is an image of my world map. This is an equirectangular projection of my world. The red grid would be a 250 miles² grid, but obviously it doesn't measure distances horizontally accurately (except at the equator) due to the distortions of the projection but the vertical lines should remain accurate, I believe.
Iota Globe Net 001 GRID.PNG

What I want to do is create more detailed maps of different (and effectively arbitrary) areas of the map, where political boarders, civilizations, and environments and such are mapped out. I want theses "zoomed-in" maps to be as accurate in terms of distance, shape, and area as possible. I'm looking for suggestions on how I might best go about this.

Say I want to map out this region here:
Crop1.png
Which would look something more like this on a globe:
crop2.png

How might you suggest I do this sort of zoom in and redraw while keeping distances and what not to scale as much as possible?

I have the following at my disposal with varying amounts of skill with each and in no particular order:
Camapign Cartographer 3+
Fractal Terrains 3
GPlates
GProjector
GIMP

I do all my final mapping (short of aesthetic changes) in CC3.

Possibly relevant numbers:
Circumference at equator: 16,000 miles
Radius: 2546.5 (rounded to the nearest tenth)
The planet is for all intents and proposes a proper sphere.

[Phergus]

Quite interested in seeing what solutions are suggested for this.

[Azélor]

You want the best map projection for each continent individually to minimize distortions?

[waldronate]

For choosing the best map projection for a given purpose, I recommend the following pages:
http://www.geo.hunter.cuny.edu/~joch...projection.htm
http://desktop.arcgis.com/en/arcmap/...projection.htm

In this case, you already have your map in the geographic projection (Equirectangular in a 2:1 ratio), so it will be easy to reproject to any of available projections provided by your tools. As you show smaller and smaller areas on the map, the distortions will be minimized for any projection, so it won't matter as much. It's the intermediate levels that will cause you problems and the first link I provided has some pretty good recommendations. Even though it's based on an Earth Map, http://projectionwizard.org/ might help you with some intuition about projections.