The Three Lions bowed out after Harry Kane’s 84th-minute penalty miss proved costly as Les Bleus progressed to the last four where they will face Morocco.
France had gone ahead from Aurelien Tchouameni’s 17th minute strike, but Kane leveled from the spot in the 54th minute after Bakayo Saka was brought down by Tchouameni.
Olivier Giroud scored the winner with a 78th-minute header, before Kane spurned his chance from the spot, firing well over the bar.
That ended England’s latest trophy bid under Southgate, who took over in 2016, guiding the side into the 2018 World Cup semi-finals and finishing as Euro 2020 runners-up.
“I think whenever I’ve finished these tournaments I’ve needed time to make correct decisions because emotionally you go through so many different feelings and the energy that it takes through these tournaments is enormous,” Southgate said.
“I want to make the right decision, whenever that is, for the team, for England, for the FA, and I’ve got to be sure whatever the decision I take is the right one.
“I think its right to take a bit of time to do that because I know in the past how my feelings have fluctuated in the immediate aftermath of tournaments.”
Southgate also revealed that Kane was feeling “low” after his penalty miss, as expected, but backed the Tottenham forward.
“He’s very low but he’s got nothing to reproach himself for,” he said. “We’re in the position we are as a team due to his goals and his leadership over a long period of time. The result was due to 100 minutes of football and things that happened at both ends of the pitch.
“Even if the penalty goes in we’ve still got a lot to do to win the game. We win and lose together.
“It was difficult when you get a second penalty against a goalkeeper you know very well (Hugo Lloris). Even the best are going to miss at times, that’s football.
“But there’s no-one I’d rather have in that situation and if we got a penalty tomorrow I’d feel the same way. It’s cruel for him because he will feel disappointed in himself, but he shouldn’t.”