What Capri said.
Often when you go taller, you also wind up with wider. The net result can be no net change, you get the gearing advantage but wind up with more rolling resistance and aero drag. You'd have to choose your new tire size carefully, you want to maintain the frontal area or, if possible, reduce it slightly. That is not easily done. Sometimes trying to find a narrower tire that works requires you get a narrower wheel too, and then things get pretty expensive pretty fast, and for a pretty small result.
When it's time for new tires, shop hard for the lowest rolling resistance you can find. I didn't get LRR tires on my HCH and that was a BIG mistake.
In the meantime, experiment with higher inflation pressures. Some guys will argue against this and they have their points, all I know is that I have noticed these things: it doesn't seem to affect my tire life at all, and I do nearly 20,000 miles per year; the ride is noticeably harsher but not so bad that it is unbearable, and it makes an immediate and easily noticeable effect on my fuel economy. I picked up, if I recall correctly, about 5% on tire pressure alone, and it cost me nothing.
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