I don't think the stage matters except if you have the objective to defeat a specific target like Juliana or Elvire.
I found this out while taking my time capturing as many enemies for sacrificial lambs for ritual points. 10 hp healing rod on Lily topping my units off before every group of enemies led to me taking over 30 turns.
The first time you take over 30 turns on a stage gives a prompt and a level 40 Lupus will appear with high stats and about 124 HP. He can be captured and his five movement along with map attack is quite strong.
The second time spawns a level 45 Evelinael with relatively high stats and a little over 100 HP. She can be captured and she only has enough SP to use her map skill 3 times. This one was the last hscene i needed.
The third time spawns a level 50 maid angel just as strong and she has a 100% chance to act twice in one turn. She spawns with the 100 HP healing consumable. She can't be captured.
The fourth time spawns a level 55 maid angel. Stronger in stats with same items and can't be captured.
The fifth time spawns a level 60 maid angel for the last time. She still can't be captured.
I tested for a sixth time and nothing spawned. I tried it for the 50th stage after defeating the maid angel three times and nothing spawned after 30 turns. I don't think there are more content beyond this.
The weapon dropped by the level 60 maid angel
View attachment 24785