I'm afraid that's how Patreon works, and it's both bad and good.
The problem is that there is a huge amount of offer (great art) for each individual to choose from, and only limited budget to support artists. Yep, it's disappointing/frustrating to get someone using a tier as a one-time-fee thing and then go. But maybe they would not have come at all if it was a contract for longer.
I'm placed at the opposite side. I only support one project on Patreon. I give them $10 which is the max I can in my financial situation. But that's since 3 years, so it starts to add up ($360). I could also have given $3 to 120 projects, or, more reasonably, $36 to 10 projects. I do feel better strongly supporting one, but I'm aware I'm neglecting others that would worth the support.
So, I think you should see things as follows:
Your $3 tier is not really a $3/month tier. It's a $3+ tier. You'll get at least $3 (or less because of fees), most likely more.
Then, under this assumption, it's a bit like buying albums on bandcamp. You can choose the minimum payment (which is sometimes a disgrace and a (self-inflicted) insult), either not caring or hoping the artist will at least appreciate the moral support, thing you generally can't know for sure; or go for a higher payment that corresponds more to the time and money investment that went into the album.
So, to come back to your issue, you have to give content in this $3 tier that is a good compromise between not disappointing the monthly supporters, and not being too disappointed yourself if some people get all of it for only $3.
Doing higher $ tiers may be an option, but it may also dissuade people even more to continue after first payment (but.. first payment is higher so less disappointing.. in theory! Because $3 or $5 do not change much things in emotional terms).