On Saturn VUEs, installing XM for friends, I have mounted the antenna at the back, and ran the wire over the rear edge of the rear door, then under the headliner, over to the right side, down and from there on, to the front under sill moldings. Then up the kick panel and to the unit.
I believe this would work for the H3.
As Sewie states, the A-pillar would be a good way too. Run the wire down the A-pillar molding, to where it ends, then back through the grommet from the door to the inside of the vehicle.
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