A 0-0 stalemate in St Etienne ensured England will be in the last 16, behind neighbours and Group B winners Wales, but there was an over-riding air of dissatisfaction.
Hodgson made six alterations, including resting his captain Wayne Rooney, but could not change what seems to be this side's one constant - a failure to turn possession and territory into goals.
They have yet to score in the first half in France, and might easily have drawn all three of their games had Daniel Sturridge not popped up at the death in Lens.
They will meet Group F's second-place side next - one of Portugal, Hungary, Iceland or Austria - and the manager's message was clear: things aren't as bad as you might think.
"We're not doomed yet," he said.
"We can't do much more, we dominated the game from start to finish, we had so many chances and one day we will put them away. I can't fault the effort and work of the players.
"It has been attack versus defence in all three games and I never thought I'd see England dominate three games like we have done.
"We will be criticised for not taking chances, I can't deny that, but I think the time will come when we will take those goal chances, and some team will be on the end of that fairly soon.
"But we'll wait and see; we're in the knockout phase which is where we wanted to be."
Hodgson appeared piqued by queries over his selection, bristling at the notion that he had rolled the dice by switching more than half of his starting XI.
While Jamie Vardy and Sturridge were widely tipped to start after their game-changing efforts from the bench against Wales, the call to stand down Rooney, Dele Alli and both full-backs seemed more bold.
"You're saying that had Wayne started, he would have scored the goals the others missed from his left half (midfield) position," he told one inquisitor.
"Wayne and Dele Alli and Harry Kane came on and created chances. The 'six changes' amuses me.
"We finished the game against Wales with Jamie Vardy and Daniel Sturridge up front and people said that was positive. Now it suddenly becomes six with those two starting."
Hodgson also faced questions for the first time about a possible penalty shootout in the knockout stages, a scenario that seems more likely every time England fail to take their chances inside 90 minutes.
He added: "They get taken every day (in training), so we'll wait and see. Penalties in training, and penalties in front of a lot of people...that's two different things as we in England know.
"We're not doomed to penalties, we're not doomed to not take our chances. If the team continues to play with the intensity and domination we've shown in these three games we're capable of winning a game in normal time."
Hodgson also defended Jack Wilshere, who took over Rooney's playmaking role but looked decidedly off colour before being hooked in the 55th minute.
"I realise his performance hasn't been too highly rated in the mass media. I've picked up on that one fairly early doors," he said.
"I think he's a very good footballer and will be a very important member of our troop and squad. If we can stay beyond the next round, I think you'll be speaking of him in a different light because that's the player he is."
Hodgson found some unexpected support in the form of Slovakia coach Jan Kozak. Twice Kozak was invited, gently, to rub salt into Hodgson's wounds and twice he declined.
On the subject of England's half-dozen changes, and whether that was tantamount to disrespect, he said: "When I watched the matches against Wales and Russia I could guess the line-up of England for today's match.
"It was a logical decision. I suppose the alternatives at full-back were very similar, the centre-backs were the same, and I guessed Rooney and Dele Alli would stay on the bench. Sturridge and Vardy scored against Wales, and Lallana stayed in the line-up. It was a logical decision of the manager."
More surprisingly, perhaps, was his response when asked to compare England with the Welsh side who beat his team 2-1 in Bordeaux.
He said: "I think that, regarding football qualities, England are better than Wales."
Chris Coleman and Gareth Bale might well differ with that assessment as they toast pipping their neighbours by a point, while Kozak must wait for results elsewhere to see if Slovakia qualify from third place.