There were politics abound in everything around them, of that Geric was more than well aware. The Tyrells were known for playing politics, and there had been concern expressed about the lord of roses and the king's safety. Geric was not one for idle rumors, but he was even less of a type to risk the king's life, in anything. There were risks of traps in the woods, who was behind it Geric didn't know. He did care, but it was not his job to find that out. He dealt with the threats they faced, as they faced them.
Lord Stark rode with as well, which Geric did not begrudge. How they had gotten along in the past didn't matter when he trusted Stark more than the other company as loyal to the king. Admittedly, if Stark were to attempt to kill the king, it was likely himself. So Geric's eyes were everywhere, his position close to the king. He was not spreading out, no. The king's life mattered most, and he stayed at a close distance to protect it.