I’ve searched it up online, and previous posts talked about other teams that I wouldn’t consider rivals, such as the Astros, which is more of a hate because of baseball in general, and not because of team rivalries. I’ve heard the A’s passed around a lot, but I’ve never felt there was any rivalry since the Angels were dominating. The two biggest I could see would be the:

Mariners – We had a big fight with them last year, but I don’t really see any rivalry nowadays.
Dodgers – “Same” city teams, and they robbed us of our best player. Any game against the Dodgers I feel this year would be pretty big.