Does this rule apply right away or after the upcoming season?

Your League performance from last season determines the multiplier with which your FA points will be calculated and determines if you’ll be eligible to claim rewards after every tournament.

If you’ve finished 1st-3rd spot in the League the season before you will be eligible to claim tournament rewards and will have x3 Multiplier
If you’ve finished in the 4th-8th spots in the League the season before you will be eligible to claim tournament rewards and will have x2 Multiplier

However in case you’ve finished below 8th spot in the League the season before you won’t be eligible to claim tournament rewards and you will have x1 Multiplier.