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.
According to solstation.com, the sun (as all main-sequence stars) slowly brightens (while remaining the same yellow-white color) through the main sequence. Even so, by the time the sun is just about ready to become a red giant it will already be 2.2 times as bright as today, meaning Mars will get as much energy as Earth does today (good argument for terraforming, perhaps?).
Estimates for the Sun's brightness as a red giant ranges from 625 times current to 5000 times current.
To calculate how far you'd need to be to have current Earth-like solar imput:
Earth Equivilant Distance (in AUs) = Square Root of Stellar Luminosity (in Solar units, so current solar = 1)
Assuming 625 times current brightness - 25 AU's, or about the distance of Uranus
Assuming 5000 times (which is, granted, probably way more than it will actually be) - 70 AUs, or between Pluto and the outer reaches of the Kupier Belt.