I'm honestly surprised I haven't written this down already. I checked through the various requests I had written, though, and there wasn't one there that was focusing on this, from what I could tell. This request is for Archer, or Heroic Spirit EMIYA, to be more involved in the storyline of Fate/Grand Order. Maybe have him be summoned after the Fuyuki singularity, and have him be recognized be Chaldea as the Archer class servant from that singularity. It makes more sense that way, in my opinion, just because it wouldn't make much sense for him to fight a version of himself that's even closer to being him than Emiya Shirou is, or just kind of is him in general. Moving on from that, though, have Archer just follow Ritsuka and Mash throughout the various singularities, and slowly reveal his past along the way, whether it be via the dream cycle or Archer himself revealing it unintentionally/intentionally. Have Archer be pre-UBW, as that would be more fitting for developing his character than if he already knew he wasn't wrong in his ideals. Make sure not to breeze over the Camelot singularity; he should have a pretty bad mental dilemma over that version of Arturia existing. Have other servants in the singularities feel like they slightly remember him from somewhere, but they cannot place it. This is due to how many times he likely saw them all throughout his various Counter Guardian missions. Have an even stronger sense of familiarity attached to ones that he met in the fifth holy grail war.
Alternatively to this, you could also have an entirely different story with Emiya Shirou, post UBW, being involved with Chaldea, replacing Ritsuka, and summoning servants that remember him from the fifth holy grail war in the singularities. These could include that war's versions of Archer (EMIYA), Lancer (Cu Chulainn), Saber (Arturia), and maybe even others like Berserker (Heracles), Caster (Medea), Rider (Medusa), or Assassin (Sasaki Kojiriou). Perhaps you could even include True Assassin, if you wanted to.