Chris Coleman's side reached the semi-finals in Wales' first appearance at a major tournament in 58 years.
Symons is set to return to the coaching set-up ahead of the opening qualifier against Moldova on 5 September.
"Because we've had such a fantastic tournament, teams will be even keener to beat us," he said.
"But this group of players that we've got, having the success we've had makes us hungrier.
"We want to qualify now for the World Cup and have another really successful tournament.
"That's what we're aiming for and I can't see any reason why that can't happen."
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Wales will also face Austria, Serbia, the Republic of Ireland and Georgia in World Cup qualifying group D.
Coleman, who succeeded the late Gary Speed in January 2012, has said the 2018 World Cup campaign will be his last as Wales manager.
The 46-year-old former Fulham, Coventry City and Real Sociedad boss agreed a two-year contract extension with the Football Association of Wales before Euro 2016.
Symons, who is set to return to Coleman's team in place of Cardiff City head coach Paul Trollope, believes his former team-mate will eventually be a success in club management.
"After the Euros and the great success we had there, he could have gone off and got a club job now," Symons told BBC Wales Sport.
"He's committed himself to Wales and is desperate to see the World Cup campaign through and hopefully it will be a successful campaign for us.
"But come the end of the campaign, without a doubt, he'll go back into club management and I'm sure he'll be very successful."
Wales will have to learn to deal with increased pressure in the 2018 World Cup qualifiers following their Euro 2016 exploits, says Kit Symons.