If you're using a tilemap, this transition is fairly easy to do. You will probably already have some sort of camera class.
The thing is you only move the camera when the player reaches the edge.
Suppose your screen is 20 tiles wide and 10 high, the camera shows 0, 0 to 19,9. If the character reaches the border of tile 19-20 set the camera to a 'transition state' now move the camera while moving your character to tile 20. So increment camera x until it reaches 20 (the tile you character is). The camera should show 20,0 to 39,9. Return the game to the normal state and play resumes.
So no need to render the entire map, just control your camera movement.
Edited to reflect the comment made below.