In terms of actual formal prerequisites for Hatcher's book, basic algebra and those notes on topology should mostly cover what you need.
If you find that the notes aren't enough, it's most likely because you don't have enough practice _using_ the theory that's presented there. The answer for that doesn't necessarily have to be another book on topology; if you're not in a rush to start Hatcher's book, the best thing to do could very well be to study a subject that constantly uses topology. For example, that could be Lebesgue integration, functional analysis or differential geometry (on manifolds).
An alternative might be a problem book in topology, as Hatcher's notes don't have a ton of exercises. An example would be the first three chapters of the book _Elementary Topology: A First Course_ by Viro, Ivanov, Netsvetaev and Kharlamov.
EDIT: Here is a link for the last book. The online version is an abridged one, omitting the proofs and solutions. <