To my understanding, Soul Echoes and Soul Fragments drop by default on Ascendant Difficulty, and their drop rates can be further increased using the altar.
The higher the monsters' tier, the more likely they are to drop them, so Red Named (Nemeses/The Dread) > Violet Named (Bosses) > Brown Named (Heroes)
Basically, we want to spawn as many hero+ monsters and kill them as efficiently as possible.
What's your strat, and how are they working so far? Havent tried SR yet.
My current one is sacrificing 3 Ashes of Awakening + 2 Frozen Asters for +100% Fragments/Echoes drop and +100% Voidmarked chest chances at the altar, then heading for places with high chest density but relatively close to portals like Depraved Sanctuary, Fort Ikon Armory, or Cronley's place (the side entrance to the mine has Totem Chance + 3 easy chests). They're usually enough for 4~5 Fragments an hour, more if the Dread decides to grace me with its presence.
Edit: It might be just me, but I observe that dangerous domains have a higher chance to spawn nemeses = more echoes/fragments.
Edit2: The dev said a while ago that they boosted the spawn rate of factions with small territories, like the feuding cults (Kymon's and Death's Vigil) and Cronley's. Seems like it affects the whole area and not just those factions, so Arkovian Docks has a very high chance of nemesis spawn includes the Bloodbound's nemesis, undead nemeses, and obviously Cronley's.
Source: r/Grimdawn · by /u/chaoton