Ahead of their first World Cup qualifier against Moldova in Cardiff on Monday, Williams says that they will not be resting on their laurels.
"It's important we don't keep going on about it but use the experience from there (Euro 2016) to move into the next campaign, knowing what's required to qualify," the defender told reporters on Sunday. "I'd use the same word as the manager in being 'desperate'.
"Having tasted it, we would love to get back there," he added. "We want that elation that we had this summer. We want that again in our careers."
Williams, who ended his eight-year stay at Swansea City when he moved to Everton in August, believes that Wales's performances in France will serve them well as they look to qualify for Russia 2018.
"We have had a couple of videos in meetings. The first one we got together and had a video about the Euros just to relive a couple of those memories and relive what it was like," the 32-year-old said.
"It was important to acknowledge what we did and it was also important to put that to bed a little bit. Use the experience of it and move on."
Wales and Moldova are joined by Austria, Serbia, Ireland and Georgia in Group D of Europe's World Cup qualifying section, with only the group winners guaranteed a spot in the tournament finals.
(Reporting by Ed Dove; Editing by Toby Davis)
