I think I finally figured it out. My guess is that you've got a movie that's actually wider than 16:9, so when you view it 'widescreen' (with the black bars), the black bars are bigger than you want and the movie is too small (top to bottom) even though it fills the screen horizontally. Is that what you've got?
This is what a 2.35:1 movie looks like on the iPhone screen:
View attachment 187311
and this is a 16:9 (which is the same as 1.778:1) video:
View attachment 187312
does this look like the problem you're trying to solve? If so, in Handbrake click the 'Picture Settings...' button, then either reduce the width of the movie in the 'Size' settings (uncheck 'Keep Aspect Ratio'), which will shrink the movie horizontally, or in the the 'Crop' settings select 'Custom' then increase the left and right numbers to cut off the sides and bring the 'shape' of the movie down to what you want.