This is what I have been doing...
When I want to download features from AGOL, I go to the hosted feature layer export it as a file geodatabase (FGDB). I then download the FGDB, which can then be opened in the ArcMap. If you look at the FGDB in the Catalog, you will see that for each feature layer that has attachments, there will be a a table and relationship class file. These are your attachments!
Then I do whatever edits I need to do in ArcMap, save the map, then share as a service, overwriting the existing service. This will replace the old hosted feature layer but retain your attachments, so long as the layers in your map are from your FGDB. I do find that, for whatever reason, if the feature layer is in a map it needs to be deleted from the map (or the map deleted) and added again for the changes to be viewed in the map.
Basically, everything needs to remain in a FGDB with the corresponding relationship class and table features in order for the attachments to stay with the features. If you import a feature class from your original FGDB to a new one, you'll notice that those two features go with it.
Now, instead of having one large FGDB with Feature Datasets, I have multiple FGDB for each hosted feature layer. For example, I do a lot of work with utilities. So I made a map in ArcMap that has multiple feature layers relating to the electric utilities. I published the map as a hosted feature layer, which has 10 or so layers within in it. So it's not like I have a FGDB for every single layer I ever use (that would be insane). Instead I have one for electric, one for stormwater, one for gas, etc and each has many layers within it. That's how I keep my maps organized on AGOL too, so it's not as messy as it sounds. I collect data out in the field, and just download the FGDB from AGOL on a regular basis (with a date in the name) so that way I always have a hard copy on my computer. See attached picture to see what I mean...
I'm sort of new to all this, so perhaps there is a better way, but this is the best way I've found thus far to keep everything attached and updated. If you lost your attachments to your feature layer, I honestly don't know how to get them back other than manually adding them back one by one. But this will keep you from losing them in the future!