How to get photo files in original size from Instagram?

Dear fellow community!

Does anyone know how to get the photo files in original size from Instagram? They surely store original files in full size, since you can upload a photo via app to your IG profile, delete it from the phone, and it still syncs your IG with phone library to get this photo back in full resolution.

Their API supports fetching up to 1080x1080px, plus I found this method via Chrome developer tools: http://ift.tt/293j8pc. Script that could do it doesn't seem reliable on large scale, so I'm still looking for a automated better solution.

Please share any experience that could help.