One of the classic combination classes, the Arcane Archer was a half-caster ranged attacker, along the same lines as the Eldritch Knight and the Arcane Trickster. Those others got Archetypes, so let's see if we can do the same for our archer friend!
The first question we have to answer is: which class gets the honor? The traditional arcane archer was a fighter/wizard, which would work today.. but that combination is already served by the Eldritch Knight. The chassis is compelling, but we don't want the archetypes to end up too similar; we'll try to build on Fighter first, and keep an eye toward staying distinct.
Spellcasting First, we will follow the same spell level progression as the EK does, but with different school restrictions, and some additions to the spell list: Instead of abjuration and evocation, your choices are mostly restricted to conjuration, transmutation, and divination spells. These spells are added to your class spell list, and available to be learned at the appropriate levels: Hunter's Mark, Hail of Thorns, Cordon of Arrows, Spike Growth, Flame Arrows, Conjure Barrage, and Freedom of Movement.
Unwavering Focus Add your Intelligence modifier to your initiative bonus. As an action, you can observe a particular enemy (that you can see) with your inner eye to learn its highest and lowest saving throw bonuses, and its CR. You must take a short or long rest before using this ability again. In addition, you have advantage on any skill check make to recall information about newly encountered foe.
Enchanted Arrow Starting at level 7, after dealing damage to an enemy with a ranged attack, you may use your bonus action to cast a spell targeting only that enemy with a cast-time of one action.
Sniper's Nest Starting at level 10, if you spend at least 1 minute preparing and observing a specific location (a distinct area up to 10 feet by 10 feet), you gain these benefits while inside it and alone: successful attacks against enemies that you can clearly see and that are unaware of your presence are automatically critical strikes, you have advantage on stealth checks to remain hidden, and you have advantage on perception checks to detect enemies.
Fleet of Foot At 15th level, you gain the ability to disengage as a bonus action, and your movement speed is increased by 10 feet.
Magic Arrow Starting at 18th level, you may as a bonus action enchant your bow until the end of your turn so that every successful ranged attack you make with it also casts magic missile without expending a spell slot - all missiles are aimed at that same target. You may only use this ability a number of times equal to your Intelligence modifier before taking a short or long rest.
My main concern was that it might be too effective - an Eldritch Knight with a bow is already one of the most effective damage-dealing builds available. To keep from eclipsing other classes, I avoided any abilities that substantially add to the Arcane Archer's ability to deal damage, focusing instead of allowing utility and mobility without interfering with that damage-dealing role.