I'm no expert but I just looked into this last week because I was worried about the same thing. Like someone mentioned, the grip is the most important part to prevent the drop in the first place. Before I suggest specific models, do you prioritize a case that covers the buttons entirely or do you prefer cutouts for a more tactile feel? I found a few solid options that fit your $40 budget:

• Spigen Tough Armor Pixel 8a Dual-Layer
• This is the standard for decent protection without adding extreme bulk. It usually sits around $20.
• Caseology Parallax Pixel 8a Geometric Design
• This has a really nice 3D texture that makes it hard to drop, and it is frequently on sale for under $18. Check Amazon warehouse deals if you want to save even more. I've found plenty of open-box cases there that are basically brand new for half the price.

Regarding what #1 said about texture, I agree—but please, just make sure you add a screen protector. I'd recommend getting it professionally done if you're not sure about the alignment.

• Spigen AlignMaster Pixel 8a Glass
• ZAGG InvisibleShield Glass Elite

Honestly this comes up a lot so you arent alone in feeling that way about the older Pixels. I switched over to a smaller form factor a while back and it was night and day for me. My current setup is basically just a thin but grippy shell because I realized I was just making my phone too heavy to actually hold onto. After dropping my old phone a dozen times I learned that the texture matters way more than the thickness of the plastic. Here is what I learned from my own drop-happy experiences:

• Texture is king; if it feels like smooth glass it is gonna fly out of your hand
• Raised bezels around the camera and screen are mandatory if you actually want to avoid that heart-stopping crack sound
• Avoid the really cheap silicone ones that attract lint like a magnet, it gets annoying real fast I ended up going with the one I got because it has these little ridges on the side that give my fingers something to grab onto. Its not a tank but it has survived a few nasty tumbles on my kitchen tile. Honestly just look for something that feels "tacky" in the hand rather than something bulky. You dont need a literal brick to keep the screen safe... just stop buying the slippery cases and you will be fine.

This comes up a lot. Before you buy, do you want a case with a built-in kickstand, or are you prioritizing thinness above all else? That changes the recs entirely...