DEM usage with Sen2Cor

I changed the “none” to “dem/srtm” to motivate Sen2Cor to use DEM during atmospheric correction.
I had 30 m resolution SRTM data (1 arc) for the area of interest and placed it in the following folder:


During the correction, Sen2Cor realized to use a DEM, but it tried to download it from the website rather than using provided files. The download process was really slow, accordingly after more than an hour I aborted the process.
I wonder if the specified path for the 30 m data is correct and if the data should be named in a specific way. The current name of the tiles in geotiff format are:


Furthermore, wonder if an image should be fully covered by the selected DEM or partial coverage for the area of interest is enough?