Jones kick-started the club's fight-back away against London Irish in round one and the last secured the all-important bonus-point against the same opposition at Murrayfield in round six to earn home advantage in the quarter-final against four-time champions Toulouse on Saturday 7 April.
Edinburgh Rugby head coach Michael Bradley said: "Lee has grasped all the opportunities presented to him improve and perform. His performances have undoubtedly lifted the team on a number of occasions and he has an exceptional engine which always helps him to hunt out scores from the opposite wing.
"Some of his highlights would certainly be his covering defensive work and an incredible strength in the contact area that regularly takes teams by surprise. Few can afford to give him a glimpse of a gap or he'll be away.
"His work rate is exceptional and he's a great professional. He brings great energy to squad and we're delighted to have such a committed player extend his time at the club."