For fighter the best non-magic is Monster Hunter, the best magic is Arcane Archer, both offer the Archery fighting style for +2 attack. MH has the best things to spend Sup. Dice on, and although Arcane Archer has some cool shots but curving arrow is awesome, so a bonus action turns a miss into a hit, you can use every round if needs be (no limit)
I have found the NEW best Magic user is Warlock, pact of the blade. At 3rd you get Improved Pact and have a +1 att/dam weapon, can have it poof into your hand at any time. You can enhance the bow (and subsequently the arrows) with magic. Being able to cast seeing 120' in the dark to shoot your enemies before they see you then cast darkness which blinds them and you can still see them clearly to shoot them (and don't let the DM cheat and allow the baddies to "know" the right way to run out of your darkness when they have now idea). Life drinker for Cha dam on top of Dex damage, and a bunch of other invocations that a creative person can make a serious advantage of.
And before anyone has an aneurysm, Improved specifically states that bows can be your blades, and you can cast the normal spells on them as if they were a blade, so Booming blade and green flame blade are perfectly acceptable spells for them by RAW.
You get an extra attack if you keep going in Warlock, or lots more attacks if you keep going in Fighter, and all three Warlock, Arcane Archer & Monster Hunter all regain their abilities after a short rest.
Rogue is a great level 1 starting dip for the skills, Expertise and the situational SA bonus, but further levels are a trap, as it only scales with Rogue and getting the SA is only situational. If you feel the need at best go 3 levels and choose scout for the mobility bonus, situational SA is now 2d6.
If you are going few warlock and lots of fighter then consider Magic Initiate (Booming, Green Flame Blade & Find Familiar) as they will scale with character gaining the bonus damage later on, and a Familiar scout is always a useful thing in this game. And Sharpshooter is also a must.
Other feats to look at are Elven Accuracy or war caster. if you go war caster and don't find a familiar interesting, then take eldritch blast (the blade spells are in both wizard and Warlock). By RAW it will mean your eldritch blast will scale by character not class, so you can multiple eldritch blast on AoO and not waste arrows/spells/Superiority Die/Arcane shots, but still hurt the bastard!
So before you reach 10th level you can shoot up to 2 +1 (or +2) magical arrows, with extra bonuses (or bonus Sup. die) to hit, Hex on the target for extra damage and disadvantage to one save and possible extra Cha damage to target and others I choose within 5' of it) spells on them for either cool effects (Arcane Shots or weapon enhancement spells) and/or lots more dice for damage (up to 2 Sup. dice), double ability damage (Dex & Cha), if you get the right situation +1d6 SA damage, and take the -5 att +10 damage, because if you roll REALLY crappy and miss, you can just curve it into the bad guy beside the one you were aiming at and odds are you roll better and hit and deliver all this cool damage.