England need two wins from their three remaining qualifiers to reach next year’s World Cup in Qatar.
Manchester City’s Phil Foden was the pick of several outstanding players in Saturday’s 5-0 win over Andorra.
“We want the squad to feel involved,” said Southgate.
Foden delivered a majestic performance for a vastly changed England in Andorra, a game in which Jack Grealish and Ben Chilwell scored their first international goals while Tammy Abraham, James Ward-Prowse and Bukayo Saka were also on the scorers list.
“It’s important that they know their worth and can contribute. It was not a friendly [against Andorra], there were qualifying points at stake and I was very pleased with the quality of the performance and the application,” added Southgate.
“I have a headache and I commit a crime every time I choose a team. We have such strong competition for places; there have been a lot of players the other day who have done their own self-confidence and reputation good and that is good news too see.
“We cannot be a team that has a day off. We want to maintain consistency, and we have had that throughout the calendar year. Everyone is proud to put on the jersey.”
England were beaten in the Euro 2020 final this summer and are entering the game against Hungary knowing that victory will bring them one step closer to the 2022 tournament in the Middle East.
They end their qualifying campaign next month, facing Albania at Wembley Stadium on November 12, followed by a trip to San Marino three days later.
“Special” is shortlisted for the Ballon d’Or – Mount
Midfielder Mason Mount came on as a substitute for Andorra 17 minutes before the end and is now in the starting line-up against Hungary.
The Chelsea player won the Champions League last season – throughout the English European Championship campaign – and was shortlisted for the Ballon d’Or award.
“It was special,” he said. “I probably found out about it at the same time as everyone else.
“To see that and to see the names, to stand next to these names, is of course a dream.
“For all the years of hard work, dedication … it shows everything pays off and it’s just the beginning, it doesn’t stop now.”
Mount says the lack of part of England’s Euro 2020 run while he had to self-isolate was a factor in his decision to get vaccinated against coronavirus.
He missed the win over the Czech Republic in England’s last group game of Euro 2020 and was an unused substitute in the win over Germany after spending time with his club colleague Billy Gilmour, who later tested positive for Covid after the 0-0 draw against Scotland.
“I am now double vaccinated,” said Mount. “Having that experience of missing the game – a top-top game – had an impact on my decision.
“It made me want to get it and get it pretty quickly.”