The source for Meade reaching the Little Round Top area is his sons account as published in Life & Letters:
Before, however, it had yet become daylight, he mounted his horse, and accompanied by Generals Howard and Hunt, and by Captain Paine, of the engineer staff, rode off to examine the lines. Riding slowly along the rear of the sleeping line of soldiers around Cemetery Hill, and along its continuation as Cemetery Ridge, and beyond, to where the land dips before it rises abruptly at the base of Little Round Top, he obtained a general knowledge of the features of the ground and of the chief accidents of its surface. As it was still dark when he had started along the lines, of course only the most salient features of the ground could be recognized. Before, however, he had finished the examination, day began to break, and he concluded it by an inspection of the right, around Chip's Hill, to the crossing of Rock Creek by the Baltimore Pike. He finally indicated on Captain Paine's sketch of the ground just gone over the position to be held by each corps, and Captain Paine thereupon, by his orders, made from the sketch, and during the morning transmitted to each corps, a tracing showing the positions."
In addition, Schultz & Mingus in The Second Day at Gettysburg give an account of Meade's morning reconnaissance. The describe how Hunt spoke to Lt. John Kinzie (Battery K, 5th U.S.) while Meade & Howard spoke to Geary near Little Round Top. Not sure if the battery was with Geary's brigades in the area that morning. They cite a letter from Hunt to Howard (dated Dec. 29, 1876). I'm not familiar with this primary source, so I can't comment on it other than what I've read in the book.
My guess is that the map was created to show where the corps were to be positioned once the entire army reached the field -- which didn't happen until late in the day. The Confederate attack began before the positions indicated were taken up. Alternatively, there was an earlier map showing the initial deployment and,this is map #2 showing a later planned deployment.