I have previously download the dem data from NASA/USGS.
https://earthexplorer.usgs.gov/ I think it was also in geotiff if you got individual files which you need to convert to hgt. Although it looks like you can get some hgt files zipped. I haven't used them since 2016.
https://e4ftl01.cr.usgs.gov/MEASURES/SRTMGL1.003/ you need to make an account to download.
For oruxmaps on my devices the 1 arc sec hgt is about 25Mb each hgt file so I stick with the 3 arc sec about 2.7Mb each file
Jonathan de Ferranti has made some corrections to the 3 arc sec SRTM
http://viewfinderpanoramas.org/Coverage ... s_org3.htm which you can download in hgt. He has been slowly adding 1 arc sec dem but mainly in Europe. I've currently got his 3 arc sec files for NZ on my phone and tablet.
Also there was meant to be a reprocessed SRTM dem by NASA (NASADEM) was meant to be released soon but I haven't been following it.