I think none can compete, none is as complete as the IPA to describe languages of the world. It has been adapted, revised, so it's also certainly the most accurate.
It's the official scientific "unit" in linguistics and phonology, like the meter/second is the international scientific unit for the speed in physics...
But: if you study only one language, and don't compare several languages, you may find another one more adapted to the language you study.
So, it depends on your goal. If you want to be understood by other people (asking if this word is pronounced this or that in their dialect) or to study comparative linguistics, IPA is the best choice (all books use it).
If you are, like I am, a wannabe polyglot, who study several languages, the best choice is to first spend a lot of time studying the IPA, because it's very well documented, and you will understand all the linguistics and phonological concepts to create your own afterwards.
So, the best is to learn the IPA, then, either:
1 create your own one afterwards. Adapting it to your needs, and you will learn a lot about the phonology of the language you study doing that.
2 use a phonetical conscript if you find one. You can adapt it, as it's not certain that this script would be as scientifically accurate than the (so much verified) IPA is.
You can try existing conscripts:
Conscripts are the equivalent of Conlangs for scripts, they are adapted or invented scripts to write a language, they are not all phonetical, but some of them are, and are often adapted to a few languages only, or only one, and you can find inspiration if you want to create your own system:
They also have resources to help you create your new font for your invented scripts:
For instance, this one is called "UA" (for universal alphabet), and is phonetical:
This one has been specially created to write French: