I suggest going up in rim size if you can. Going up in tire size can affect the maneuvering dynamics of your pilot and it will throw off your speedometer. Going up in rim size will give you a fuller appearance in the wheel well. Just look at the pics of my pilot. I went from the OEM 18 inch rims to 20 inch rims, but kept the outside diameter of the tire the same. It looks like it is lowered even though it is not.
Our Pilot is, um, 'sluggish' in the giant scheme of things, so adding taller tires would need to be carefully considered at least in our case. They slow acceleration and offer a larger mass/moment to get turning or slow down. The brakes in the Pilot are less than stellar too IMO, no need to reduce the effectiveness further with larger diameter tires.
There are times when I think 'longer legs' would be nice for fuel economy, but that would only help with level-ground cruising. We just don't do enough of that. Plus we are at 4000ft altitude, so somewhat torque disadvantaged already; taller tires would make that even worse.
There are folks here who have installed larger Geolander G05 offroad-capable tires on Pilots. It would be interesting to get their perspective and experience, at least their on-road impressions.