Most people will suggest using the mate with most recessive fur you have available. I often recommend starting with a mid-level fur, though.
The idea is this: say we have 40 furs. Your starter will have one of them. All things being equal (which we do NOT know), it's about 50/50 that the starter will hide one of the 20 more dominant furs, or one of the 20 more recessive. So, if we start with something in the middle, and it's one of the more dominant furs, we're not 'wasting' a breeding cycle for our most-recessive furs. A retired mid-level fur is best for this. Since it's retired, it is impossible for the starter (unless it's an old one you found in inventory) to hide the retired fur.
So, you have three possible outcome:
1) You get the starter's shown fur. This box will hide either the shown or hidden from the other mate. It cannot advance you to your goal, so, while the box may be good for other things, I classify it a failure to be discarded (sent to the Menagerie). Try again. The odds of this result are 1-in-2.
2) You get the mate's shown fur. This box is guaranteed to hide the starter's hidden fur. If it can mate with the starter, open it and, next week, when it starts gaining love, mate it with the starter parent. If you get the wrong gender, try again. The odds of this result are either 0 (impossible), 1-in-2, or 1-in-4.
3) You get a fur which is neither parent's shown fur. This might be success. Check Saga's charts. If the box shows a fur which is dominant to the mate, it is the starter's hidden fur trait and you have succeeded. If it is not, it is functionally identical to result (2) above. The odds of this result are either 0 (impossible), 1-in-2, or 1-in-4. If the box shows a fur which is recessive to the mate, the starter hides something even more recessive. The current pair can NEVER show the starter's hidden; switch to the offspring box, replacing your original mate-parent, as with case 2.
Assuming you've discarded all case (1) results, and you did not get a case (3) success, once you get a box which is the gender needed to breed back to the starter, switch to it. You now have a 1-in-4 chance of obtaining an offspring showing.
The odds for cases 2 and 3 depend upon the mate you chose. If you choose a more dominant mate, you'll get case 2 more often. If your chosen mate also hides something which is also dominant to your starter's hidden, it makes a case 3 immediate success impossible.
This is why people often recommend using your most-recessive mate. It makes case 3 more likely. Usually (unless your starter hides a new most-recessive-of-all, which is unlikely) it make case 2 impossible, and a case 3 immediate-success can occur.
The reason I recommend starting at the mid-level is usually you have something more important your deeply-recessive traits can be doing (pulling out hiddens from non-starters, crossing traits together, etc.) In general, across a large number of breeders and outcomes, it will work just as well to use a mid-level trait as a deeply recessive one. In fact, if you do happen to be poised to discover a new most-recessive-of-all, which mate you chose is completely irrelevant, all work equally well (er, equally badly
).
KittyCatS lover and LazyCats auctioneer. 
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