52-card trick for a larger deck?

Long ago someone demonstrated the following card trick with a standard 52-card deck:

(1) A volunteer selects 5 cards from a shuffled deck, which the performer does not see.

(2) The assistant puts exactly 4 of those in a pile and passes it to the performer.

(3) The performer can then identify the 5th card.

There are a few ways to execute this trick, one of which goes as follows.

(1) The assistant finds a suit common to 2 of the 5 cards, with values $x$ and $y$ modulo $13$.

(2) He then removes one of them, say $x$, such that $x-y \in [1..6]$ modulo $13$.

(3) He then puts the other one at the top of the pile.

(4) And he puts the 3 other cards after that in a permutation corresponding to $x-y$ modulo $13$.

Since (1) works by pigeonhole, (2) works because $\frac{13-1}{2} = 6$, and (4) works because there are 6 possible permutations of 3 cards, the trick succeeds. My question is whether the same trick can work with more than 52 cards. It is not obvious to me how to do so, nor compute the maximum number of (pairwise distinct) cards possible. Also, it seems to me that even if the trick can be made to work for $n$ cards, it may not work for $n-1$ cards! Any ideas?

For the $5$-card trick, a $124$-card deck is the maximum. Papers by Michael Kleber, Colm Mulcahy, and Shai Simonson and Tara Holm are good places to look for explanations, as well as some history of the trick.