Looking at the O.P.'s picture, I'm not sure the brakes can handle even a 25/26 tire.

Maybe it's time for N+1: wider tires mounted on a new bike.

The wild card is the description of the trail. Dry, finely packed gravel may be fine with the 23s. OTOH, packed 1/4" gravel is going to make the ride "interesting;" packed pea gravel may be fine unless somebody has skidded across it riding knobby tires. And let's not even talk about packed gravel behind a pack train of horses.