Where does “hallway” come from?

I notice the chain provided doesn't connect to "hallway" — it traces "way" and "away" back through Proto-Indo-European roots. Without a valid etymological chain for "hallway" itself, I cannot write an accurate origin sentence. Could you provide the correct chain for "hallway"? Typically it would involve "hall" + "way", requiring separate chains for each element.

hallway (English): A corridor in a building that connects rooms

Definitions

  1. A corridor in a building that connects rooms

Ancestry of “hallway”, step by step

hallway traces back through two separate lines of ancestry.

via English Hall

StepLanguageWordMeaning
1EnglishHallHall class, a class of steam locomotive used on...
2Middle Englishhallehall; Alternative form of hale; Alternative form...
3Old Englishhealla hall; residence; a large room forming part of a...
4Frankishhallularge room, hall; hall
5Finnishhaludesire; urge, itch
6Finnishhalatato hug, embrace; to want
7Proto-Germanichalōnąto call, appoint, summon; to fetch, get
8Proto-Indo-Europeanḱelh₁-to warm up, to be hot; to warm up, be hot

via English Way

StepLanguageWordMeaning
1EnglishWayChristianity
2Middle Englishway
3Old Englishweġway; path; way, road, path
4Proto-West Germanicwegway, road
5Proto-Germanicwēgazwave; sea
6Proto-Indo-Europeanweǵʰ-to bring; to transport
Every word from Proto-Indo-European ḱelh₁-