Perhaps, but it will still get too hot. If you take a star like Arcturus, which is about as luminous as the Sun will be as a red giant (Arcturus was probably similar to, or slightly bigger and brighter than the Sun in it's youth), you find that the area where a planet would be comfortable is around the distance of Uranus or so. So if we moved Earth, eventually we'd crash it into Mars or Jupiter, or at least run a major danger as we cross other planets' orbits.
If we moved the earth there would be no reason whatsoever to cross the orbits of other planets. First of all, the planets each orbit at their own unique incline. In order to pass the actual orbit of another planet, we'd have to do it on purpose. Secondly, even in the minuscule chance that we did cross the actual orbit of another planet (the orbit is a ring, not a sphere), there don't see why it'd have to be during the short time that the other planet is at the cross-point, rather than, say, the other side of the sun. But that's irrelevant, because there's an absurdly low chance in the above scenario that we'd move the earth into another planet's incline anyhow. If you were looking at the solar system from the side, the earth would be passing well below or above wherever Mars or Jupiter was orbiting.
And the location that we'd need at the peak of the red giant stage is approximately 2/3 between the current location of Mars and Jupiter. And as I mentioned, the current consensus is that earth will move in any event, and so will the other planets since they're of course subject to the same fluctuations in the sun's gravity.
PS. And if the location we need turns out to be where Uranus is today, that's no big deal if whatever civilization is here can move the earth. After all, we've got at least a few hundred million years to work on it. Moreover, we should have the terraformed Venus and Mars to work with as well. I guess they'll need moving too.
And there's other big fish to fry before then BTW! Such as the more intense solar waves ripping away the earth's atmosphere. We'll wanna do something about all that....
As long as Uranus stays a comfortable distance from me, we'll be fine.