Bale missed Wales' best opportunity of the game, firing tamely at Hungary goalkeeper Peter Gulacsi when the ball arrived at his feet 12 yards from goal, and Mate Patkai punished his wastefulness by scoring the winner 10 minutes from time.
After opting to play Dan James as a lone striker while Bale started on the left wing, Giggs said the forward's morale was not a problem during training despite the 29-year-old having endured a difficult season in LaLiga.
"He is disappointed like all of us with the two games," Giggs said.
"I have said before, in the training camp he was great, about the place he has been great. It's difficult fitness wise having not played for a long time and then that short turnaround.
"The way he plays as well, those high-intensity sprints. It's difficult. But he's been fine around the place, no problem.
"It's always a bit of a juggling act trying to find where Gareth can be effective. We have so many players who can play off the right or number 10.
"Gareth has shown over the last few years he can play up front, number 10, off the right. It's my job to try and get his best position and to get the other players in their best positions as well."
The defeat was Wales' second in four days and left them second-bottom of Euro 2020 qualifying Group E after three games played, and Giggs acknowledged his side face a struggle to get their campaign back on track in their remaining five matches.
Group E strugglers Azerbaijan are Wales' next opponents on September 6, by which time Giggs will hope his squad is in better shape.
Having seen his side lose 2-1 in Croatia three days before their reverse in Hungary, Giggs said: "We've had two tough away games five weeks after the season finished, and you saw it in both games - a lot of the players were short of match fitness and we're asking miracles of our players.
"Ethan [Ampadu] has not really played, I did well to get 50 minutes out of him. Brooksy [David Brooks] was the same. Asking Gareth [Bale] to play two games in a short space of time with the way he plays after not playing for six weeks.
"But still, we had enough chances to win the games. It's tough, not enough of our players were at peak fitness but we have to probably win nearly every game now.
"We've made it really difficult for ourselves and we're probably relying on other teams to drop points, which they will do, because it's a competitive group.
"But obviously we have to start winning, if not win all of the games, then the majority of them."