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The TV antenna should improve the FM reception unless they've blocked the FM band to prevent interference with the adjacent TV channels. A passive splitter (cheap) might give you a better FM signal but could make weak TV signals worse. In that case, and amplified splitter (somewhat more expensive) would be a better bet; larger and needs access to ac, though.
A car antenna might be the simpler way to go, but they need to be mounted on a metal surface to work best.
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