When employing [ordinal numbers](http://en.wikipedia.org/wiki/Ordinal_number) you are explaining a relationship between items in a set.  The term you use to describe those numbers will largely depend upon the items in the set.

For example:  Primary care vs. Tertiary care.  You would describe these levels as tiers.  Hence, if there were quaternary care, etc. you would describe this as higher-tier.

If you are speaking of classes:  First class, second class, etc.  You would call these higher classed (and of course these would typically go in reverse order first being highest).

If you are speaking of orders:  Tertiary, Quaternary, etc.  You would call these higher-ordered.

And, in any case, if you wish to describe a portion of a set, you would typically define your terms:  *e.g.* Higher-ordered thingamajigs (Quaternary and beyond).

Of course, given that *order* is within the base [definition](http://www.merriam-webster.com/dictionary/ordinal) and [etymology](http://www.etymonline.com/index.php?term=ordinal&allowed_in_frame=0) of ordinal, I would say high(er)-order is always superficially correct.