I realized because I keep telling people to read The Spectre (2001) I should maybe provide a reading order in case you want to read it with context for Hal's arc. You definitely do not have to read all these comics to read The Spectre! as with all comics you should pick what sounds good to you. but they do provide context for where he's coming from and I think his arc as a whole is really satisfying.
League of Comics Geek list
Bold is recommended, italics is bonus.
Emerald Dawn #1-6 - Hal's post-crisis origins. He gets the ring, he crashes a plane, he goes to jail (not in that order.)
Green Lantern 1990 #48-50 - Heel turn! After the destruction of his hometown Coast City, Hal tries to use his power to undo it and destroys the planet Oa, his bosses the Guardians, and most of the Green Lantern Corps.
Zero Hour #0 - Hal tries to destroy all of time to set things right. This is a big crossover event spanning multiple universes - do not worry about who everyone is.
Green Lantern 1990 #0, 62-64 - Following the events of Zero Hour, Hal fights his successor Kyle in an attempt to get his ring back.
Parallax: Emerald Night, Final Night #4 - Hal sacrifices himself to destroy a creature eating the sun. If you pick one of these to read pick Emerald Night, which has the meaty character stuff.
Green Lantern: Legacy: Last Will and Testament of Hal Jordan - This doesn't add much to a Spectre read but it follows off of the Parallax arc so if you're already at it you might as well pick up this side story where Tom Kalmaku, Hal's mechanic and close friend, grapples with Hal's betrayal. I just think it's really nice to see Tom get to be a fully realized character with his own complex emotions about Hal and not just a sidekick.
Hal Jordan as the Spectre:
Day of Judgement 2-4 - You reeeally don't need to read Day of Judgment and I did not enjoy most of it but I'm including it for completionism's sake. All you need to know is that Hal's soul is summoned from purgatory to inhabit the Spectre, the embodiment of divine wrath, after its previous host Jim Corrigan leaves.
**Legends of the DC Universe #33-36 - Hal's early days as the Spectre dealing with the fallout of Parallax. If you skip everything else on this list I'd still recommend these four to get you up to speed.
**The Spectre 2001 #1-27 - 1-14 are Hal settling into what being the Spectre means for him, 15-27 deal more with his niece Helen and space adventures. Spectre 2001 has never been collected in trades and is not on DC Infinite so you're going to have to go with scans for this one but the scans do collect the letters pages which are a fun read. I miss letters pages.
Arcs and stand-alones if you just want to dip your toes in:
#2-4 - Redeeming the Demon - Hal wrestles with what it means to be the Spectre. Also, Superman and Batman are here, and so is an all-powerful woman from outside of time.
#5 - The Two Faces of Evil - If you like Two-Face this is a comic for you. I don't know where I come down on this as a story about mental illness but it's interesting!
#6-9 - The Redeemer/Gates of Heaven, Doors of Hell - these are the most explicitly religious of the bunch so ymmv BUT they're the first appearance of premium weirdo Stigmonus and also Helen's parents die in these so they're pretty important for her.
#10 - Joker's Last Laugh tie-in - even if you hate the Joker, this is stunningly drawn and is in fact a really good Parallax issue.
#11 - The Mission - good encapsulation of where Hal's ended up as the Spectre. Certainly the most sympathetic portrayal of terrorism you're going to see in a Big Two comic that came out in January 2002.
#14 - The Silver Room - just a really beautiful vampire story.
#15-17 - Mystery in Space! First appearance of Materna Minxx, Helen's cosmic nanny.
#20 - My Perfect Life - Hal helps a man with a seemingly untouchable existence.
#21-23 - Stigmonus and Son - Hal Jordan gets his own perfect life back. Mostly including this because a) I like it as a story arc when it's what Parallax was trying to accomplish the whole time and b) One Of Hal's Spectre Aspects Is A Woman. Yay :)
#24 - They Say It's Your Birthday - Helen birthday issue! Perfect incredibly sweet issue imo.
#27 - The Beginning of the Beginning - Wraps up Hal's arc with Helen beautifully.