Decent info above but sketchy conclusion. Don't replace your amp to match cheap low end speakers. $40 speakers are not worth the can of worms this opens. This is just a friendly caution after reading ideas like this for years and reading how it usually turns out. This is not your grandma's 97 Buick. You have to really understand car audio and the electronics of this car to avoid a serious case of Malibu Diarrhea.
Yep, you're right, I don't know the specifics of this car. And It's my fault for not asking. If the system is OEM and was functioning correctly prior to the upgrade, my conclusion still stands.
It sounds like you may know or at least be aware of the following but, I'm putting it out there for others that may not know.
In this specific case IF it's an OEM system and was working well before the speakers were upgraded, a High to Low level adapter would be used to take the speaker level outputs and convert them to a preamp (low level) output for a micro amp like Alpine or Sony. The amp would be wired in, then hidden behind or under the glove box and viola, done. Not even close to acceptable for an audiophile but a significant upgrade for most casual listeners. Those cheap speakers will sound fine with an upgraded amp that resolves any impedance or underpower issues which are the two most likely reasons the current speaker upgrade was a disappointment.
I left out a ton of information because car audio for a casual listener can be a rabbit hole.
8 out of 10 times, just a speaker replacement ends up in disappointment for just the reason I mentioned. The OEM built the system to work together adequately. Note I said adequately. When you replace one part in the chain (usually speakers), the system may not respond (sound) how it's expected to for the basic reason already covered. Impedance is the bigger one these days followed by underpowered.. Other things include but are not limited to sensitivity, speaker design for the space its going into, frequency response, crossover roll-off, etc, etc, etc. I can get as detailed as you think you want or think you need.
For most casual listeners, a simple speaker and amp upgrade is all they need to make them happy. The rabbit hole is simply them realizing that when the sound does get better, they want more...either higher volume or crisper highs with clearer lows, or whatever they end up realizing they want.
There is no "amp" to replace in the basic system. It's part of the Head Unit. I'm not even sure the Bose system has a separate amp. Either way, an amp can be added to the base system. It won't give the perfect sound but it's a simple solution that won't require getting into possible programming and functionality issues of replacing the Head Unit which is built into the driver interface (Touch Screen).