The Chiefs, who had the third pick, made it known to everyone before the 1988 NFL Draft that they intended to take Smith. The Detroit Lions, picking second, threatened to pick Smith and the Chiefs were forced to move up one slot to make sure that Smith would be their pick. Incidentally, one of the draft picks the Chiefs surrendered in order to move up turned out to be star linebacker Chris Spielman. Smith's pre-draft measurables were head-turning. He was 6'4½", weighed 260 pounds, had a 7-foot-1½-inch arm span, and ran a 4.55 forty-yard dash.
Smith finished his 13 NFL seasons with 105 sacks, 30 forced fumbles, 12 fumble recoveries, 19 return yards, and 1 touchdown. He also intercepted 4 passes, returning them for 68 yards and a touchdown. He is the former co-owner of the Kansas City Brigade of the Arena Football League.
On October 22, 2006, Smith was inducted to the Chiefs's Hall of Fame.