I vaguely recall that he didnt have much time to get the message delivered and that was the shortest way to get there.Running on the left vs. running on the right is the same... except for this fact that wasn't shown in the movie, which makes sense why he went left:
In real trenches the back would of had a high support to help camouflage soldiers peeking over. This would of made it very hard to climb over where as the front is designed to be climbed quickly.
From this picture, he went left and ran while the soldiers ran towards the enemy (also on the left), with the Germans shooting at them (on the left). Why didn't he just go on the right, so that no one will be in his way while he makes that run for it?Do you not remember the whole point of the movie? He was trying to