Hodgson left the post he has filled for four years immediately after last night's defeat by Iceland, and FA chief executive Martin Glenn, vice chairman David Gill and technical director Dan Ashworth will work together to find a replacement.
"Anybody who is out there who has played football will have some idea of how I feel today," he said.
"Martin Glenn, Dan Ashworth and David Gill know that if I can be of any assistance in any way then I will do so but it is up to them.
“We are still recovering and I am very fragile. It was not a good night for anyone”
Roy Hodgson
"I am now an ex-England manager and I can only wish the next England manager the very best of luck. I am leaving behind me a good group of players to work with."
Hodgson added: "Nothing in the first three games gave me any indication we were playing as poorly as we did last night, but we did.
"If you don't give the performance you are capable of, even against a team like Iceland who were well organised and played well on the night, they can beat you.
"That is what happened, we are still recovering from that and I am very fragile today. It wasn't a good night for everybody.
"We wanted to do well and we even believed that if we got to the quarter-final we could go beyond that."
Roy Hodgson and Martin Glenn speak to the media following England's Euro 2016 exit
