I feel like increasing the base number of attunement slots- or for that matter decreasing it- is a great way to tweak the feel of a given campaign.
Personally, I like the way the three slot limit works, and I've got a couple of ways that I've been tweaking it via how I design or convert magic items.
The first is to give real thought to whether a given item should require attunement at all. I'm actually in the middle of a major revision of my magic items right now and one thing I've found is that now, years into running 5e, I'm erasing "requires attunement" from about 25% of the items that I originally gave it to.
The second is to address attunement requirements within specific magic items themselves. I was converting the magic items from Adventurers Vault 2 (4e), and within it are a bunch of item sets- for example, the "Raiment of the Phoenix" item set includes a cloak, crown, mask, and scepter, and you get extra powers out of them the more pieces of the set you have. I gave these item sets the following special trait: "You only need one attunement slot to attune to this set of items, in whole or in part. However, while it is split up, different creatures can attune to its different pieces."
I also converted a bunch of items from the 3e Magic Item Compendium that are crystals that you attach to weapons or armor to give them extra powers. For these, I added this trait: "If you clip it onto a weapon [or armor, depending on the item], you can attune to both the weapon and the crystal with the same attunement slot."
I also have a few items that modify the attunement rules, for instance:
Ring of Rings
Ring, legendary (requires attunement)
While you wear this magic ring, you can wear one magic ring on each other finger without needing to attune to them, even if they normally require attunement.
(This is a conversion of a magic ring I used in 2e to make a villain modeled after the Mandarin.)