Quote from Limerent
I currently use Manga Rock on my phone. You can download manga to your device if you have the paid version. In addition, I think they have some kind of premium currency called "Rocks" that you can use to download a limited amount of manga. The free version just lets you read them online ...
Thanks for the info!
I was also thinking about looking outside the specific manga-reading apps to see if there were some general comic readers that would work. I haven't really looked in to them yet, but I do remember seeing some ads for comic readers that seem like they would work fine for manga as well.
If I'm not able to download manga directly though the app, it would also be fine if I could load the files on myself and then read them afterwords. Not quite as convenient, but it would work. As I said, Perfect Viewer on Android was perfect for me, but I haven't yet found an exact equivalent on IOS.