You’re probably aware of the fact that the level of detail of the output (the downloaded map) depends on the scale of the map. Maps4News tries to generalise towards the size and scale to prevent too large output.
In order to get full detail of the complete area, increase the size of the map, for instance even up to 1.5x1.5 meter (60x60 inches) AND leave the scale on a small level. Note that computing the map may take some time and make sure to save the project.
The result will be larger than normal, but you will get the most detail in your output for you to use in Illustrator, where you can re-scale your map to your needs without losing the required details.