Can I get a tenure track Applied Math assistant professor position in a top US college if my PhD was in a similar ranked private U.S. college? I have almost completed my Ph.D. in applied mathematics at a highly ranked private U.S. college. I am an older student with almost a decade of industrial work experience between undergrad and grad school, and so I would prefer not to go through a postdoc before I can get my teaching career into motion. As for publications, I have two (maybe three by the time I graduate) first-author papers. I have some teaching experience as a TA in the first couple of years, and I also taught my own course here last semester. My career interests mostly lie in the form of teaching, but I would also like to conduct some good research.

I have observed hiring trends and it appears as though it is impossible nowadays (without a postdoc) to get a tenure-track position at any college of higher rank than the place you got your Ph.D. (The rank doesn’t matter that much to me, but I understand the correlation between such rankings and the quality of research (and research funding) that will be available to the assistant professor). The alumni from my program, who are in tenure-track positions, are all in lower-ranked colleges – except for one holding a postdoc position in a college of similar rank.

Would you say I probably have to choose between doing a postdoc at a higher-ranked school (then apply for TT jobs at such higher-ranked schools) or simply return to industry?
If you want to get your "teaching career" going, I wouldn't worry a lot about the ranking of the school. Teaching focused positions are going to look at your teaching experience, if you don't have a lot of this then you may need to take a visiting position to get some experience. 

A school ranked lower than the one where you got your PhD might view the name of your school as something they want associated with your department. Or they may assume you are more interested in research and not with teaching. These positions can still be incredibly competitive, but if you have the right background you may have a chance to be hired right out of your PhD.

If the school you attended was highly ranked, then schools ranked the same or higher are going to want most of their tenure track faculty to be focused on research, or at least to have a good research program to get undergraduates involved in. If you don't have a stellar publication record, you will absolutely need a postdoc. Even then, you will need a good postdoc with good research results. Even then, it will be very competitive.