As you mentioned, Carole Hillenbrand, The Crusades: Islamic Perspectives (Routledge, 1999) is a very good place to start. It's very thick though, just in the sense that it has a ton of information, which can be a bit overwhelming sometimes. Some other excellent starting points are:

Niall Christie, Muslims and Crusaders: Christianity's Wars in the Middle East, 1095-1382, from the Islamic Sources (Routledge, 2014) - this one is a much shorter summary, perhaps an easier entry point than Hillenbrand. Like Hillenbrand, it also has a lot of selections from medieval Islamic sources.

Paul Cobb, The Race for Paradise: an Islamic History of the Crusades (Oxford University Press, 2014) - this is longer than Christie's book and more of a regular historical narrative, rather than a collection of excerpts from medieval sources. It's also more up to date than Hillenbrand, so this would also be an excellent place to start.

Angeliki E. Laoui and Roy Parviz Mottahedeh, eds., The Crusades from the Perspective of Byzantium and the Muslim World (Dumbarton Oaks, 2001) - this is an edited volume with chapters by different authors, and some of them are about the Greek perspective, but it is still very interesting and useful for Muslim perspectives.

There are some other collections of medieval sources too:

Amin Maalouf, The Crusades Through Arab Eyes (Shocken Books, 1984) - this is often mentioned in discussions about the Muslim perspective, and the book is very popular, and maybe even more importantly, widely available in reuglar bookstores. Maalouf isn't really an historian though, so personally, I don't really use it.

Francesco Gabriele, Arab Historians of the Crusades, trans. E. J. Costello (University of California Press, 1969) - this is basically similar to Maalouf's book, but something that is used much more often by historians, while non-historians might enjoy Maalouf better.