So, I have been playing for about 4 to 5 months now, and I can relate to the general frustration everyone has with regards to 3* and 4* ascension drops. A number of great ideas on how to increase those drops have been suggested, many of which, however, would likely require fairly large-scale changes from the Devs (entire new mission categories, etc). The answer, as I see it, is already in the game in the form of Rare Quests.
Most of us playing the game, and RPG fans in general, are goal-oriented people. We want to rise up to meet the challenges required to move forward; sometimes those challenges can be met quickly, sometimes they take a lot of time, thought, and dedication, but regardless, we feel a need to know what it is going to take to achieve the next level of success or at the very least that it is up to US and not some random computer quirk. This is what makes the 3* and 4* drops frustrating, it’s not really a “challenge” at all. Going through the motions and just praying the probability gods drop something in your lap so you can move forward is counter-ethical to what makes us fans of RPGs at the base level.
Therefore, my suggestion would be to keep the drops the same, you can’t have every casual player throwing down fully ascended 5s. But make the Rare Quests, where there are 5 levels, each progressively harder, an every 6 day feature. Rotate the elements each time, but if you meet the “hard” challenge each week you are guaranteed a 3 ascension item. Meet the “very hard” challenge and you are guaranteed a 4*. At a standard elemental rotation, that’s one of each a month. If I’ve been playing forever, and all I need is that 1 warm cape, it might be 3 weeks away, but at least I know when it’s coming and I can plan my gameplay around it. And by all means, MAKE IT A CHALLENGE. No giving stuff away, you should have to work to get them, but I honestly believe players will go all-out to get these items on a weekly basis. If I’m 200 power short of the 4* item, but I reaaaallly need it, I’m bringing a full suite of battle items in and going 100% for it.