I was thinking they need to start recording and widely publicizing exit polls using the approval system so that people can get a sense of what the alternative 'feels' like. The big problem is that it'd take at least ~three election cycles for most people to appreciate the difference it'd make.
E.g., in the first cycle, extreme candidate A wins the actual election while the moderate, B, is shown to be widely tolerated under the approval system. Nearly half the voters--those who voted for the opposite extreme, C, end up loathing the results of the actual election and wishing in retrospect that the approval system had been in place. Perhaps in the second election cycle, candidate A wins re-election because they're a known-quantity/incumbent.
Finally, in the third cycle, history starts repeating itself and people get tired of A's party. E.g., the Overton window slides back towards C's perspectives leading to C being seated. This time the people who still prefer A are disgusted with the outcome and prefer the outcome of the approval system.
At this point, half the voters have long argued that the approval system makes more sense and the other half has also come around to thinking maybe it could bolster their position as well. It'd only remain for less partisan activists to e.g., get a petition going towards a referendum.