Secondary Skills for an Archer to focus on?

If you can get the arcane smithing perk you can modify unique weapons like the nightengale bow. This will allow you to do more damage with the gear they have instead of relying on enchanting your own ridiculous bows.

Alchemy is always good to dabble in because the ingredients are everywhere, and making any potion enhances the skill. Traders always restock their ingredients every 48 in game hours. This means you can fast travel to riften then markarth and every fast travel will have a restock as its a 25 in game hour journey from one to the other. However alchemy can be tedious, and there are already plenty of helpful potions that fortify skills already in the game so making them isn't completely necessary unless you want to "break" the game.

Enchanting is always good because you can fortify archery on bracers thus making your bows do considerably more damage. If you wish to take this tree you may as well go the distance and get every perk. There is an abundance of soul gems in the game, the college in winterhold constantly restocks as do any magic shop and if you can get the daedric artifact Azura's Star (an unlimited use soul gem) you won't have any problem getting the skill up. Like alchemy, enchantment can be exploited to "break" the game but this takes some work and the game is plenty fun without the exploit.

My personal recomendation is level all 3 a bit, see which one you like doing more and then focusing on that tree. Smithing can get you the most all around increase as it doesn't add elemental damage which some enemies resist. Enchanting is better if you just want to supplement your skill in general without having to go back and improve the weapon itself every time you find a new shiny bow. Also the fact you can dual enchant weapons to have multiple effects makes it the other viable choice. So it really comes down to if you want to use enhanced effects or base equipment bonuses.