A good History of Math course would probably be enjoyable, and give you a good idea of what things are and where they lie.
From the mathematical point of view, Linear algebra is often a good "first advanced mathematics course", a good jumping-off point: it is still concrete enough that you won't get lost in a sea of abstraction (a possible issue with abstract algebra depending on how it is taught), cover entirely new ideas ("advanced calculus" can feel like you are just re-treading the same ground you already know, and depending on the specific topics analysis might also feel like it), but it should make you work through proofs and concepts in a way with which you probably have not done so far. In addition, linear methods will show up all over the place later on, so it would prove useful.
In that same vein, Number Theory can be a really good "first abstract course in mathematics", while sticking close to things you are very familiar with (the integers and rationals) while also probably delivering some exciting surprises. It often surprises a lot of people just how much of mathematics arises out of number theory (complex analysis and abstract algebra, to name just two).
If you want to stick to the applied side, differential equations is a good place to go as well. Linear algebra would be useful there, though.
So, I would suggest linear algebra or number theory first (if you can also get a good history of math course, do that as well), then decide if you want to go towards abstraction (in which case, head to abstract algebra, mathematical analysis, or whichever of linear algebra or number theory you did not take) or more towards applications (differential equations, a good advanced probability/statistics course, or a discrete mathematics course).
It is indeed the case that mathematics starts branching out, but some of the most interesting things happen where the branches meet; it would be ideal to be able to take a good one semester or one year sequence in the major areas (analysis, algebra, differential equations, topology, logic/set theory, number theory), then go on to more advanced courses in whichever area(s) you find interesting. But the truth is that this is very hard to do: not only would such a wide choice not be available except in the largest universities, but it would also mean a lot of your time. I did my undergraduate in Mexico, where all I did in college was mathematics courses, and it basically took six semesters before that had been covered (in addition to the calculus sequence, a linear algebra sequence, an abstract algebra sequence, a mathematical analysis sequence, differential equations, discrete mathematics, complex analysis, probability and statistics, plus some other stuff to "fill in the corners"; it would be barely possible to do it in two years if you are not taking the calculus sequence, but not if you are also taking other coursework as you would be in most institutions in the United States).
I've seen the book you mention before, but it's been over a year, so I don't recall exactly the nature of the material you're seeking - rare stuff not in other books, or material covered in books like Rudin/Ross?
If you're looking for creative/nonstandard "Putnam" questions over the standard material (which I believe is the nature of the book you mention), try the three volume "Problems in Mathematical Analysis" series by Kaczor and Nowak. Lang's "Undergraduate Analysis" (with accompanying solution manual by Shakarchi) also covers some "beyond the box" material.
Best Answer
You might keep the well known graduate analysis textbooks by Folland and Royden in mind, but it looks like they are out of your price range. You might be more interested in Stein and Shakharchi, though I don't own it.
I own Lieb and Loss's book and I don't think it is appropriate for a student just finished with undergrad analysis--more like a second semester graduate textbook since it really deals with functional analysis. I also own Knapp. It will contain a review of advanced calc in the beginning and then you'll learn pretty much everything covered in a first semester graduate analysis course.
But as MSRoris said, doing little Rudin thoroughly is definitely worthwhile. If you want to be serious about this stuff, take the grad student approach and do every problem!