Question: Last Shabbat I wore a suit that I had not worn in a while. On Shabbat morning I happened to check an inside pocket and found a $20 bill inside. Upon making that discovery, what should I have done?

Answer: There are few questions one has to ask. While the money is muktzeh, does it make the suit jacket muktzeh? Even if the jacket is not muktzeh, may one continue to carry around the money?
The general subject of this question is of basiss l’davar ha’asur, an object which is intrinsically not muktzeh but it is supporting something that is muktzeh. The basic halacha is that the basiss is muktzeh and that if this situation existed when Shabbat began (at first glance, your case), it cannot be moved even after the muktzeh is removed (Shulchan Aruch, Orach Chayim 310:7). However, there are many exceptions to the rule. You did not provide much detail, but presumably at least one of the exceptions applies, which would have partially solved your problem.
When two intrinsically separate objects are firmly attached, one of them is the main part of the joint object, and the less important part is a basiss l’davar ha’asur, the joint object as a whole is not muktzeh (Rama ad loc., based on Shabbat 44b). This is the case regarding pockets that are sewed onto an article of clothing, where the suit, in your case, is more important than the pocket. Therefore, if the pocket has money in it, the suit is not considered a basiss to the money or to the pocket, and the suit can be moved. Even when the garment is not muktzeh through the laws of basiss l’davar ha’asur, the pocket can become a basiss. In that case, one should not stick his hand into the pocket (Mishna Berura 310:29), even in order to remove the money (Orchot Shabbat 19:302) and even after the money has been removed.
There is a type of pocket where the whole garment can become a basiss l’davar ha’asur. When a pocket is not just sewed onto the fabric but the fabric serves as part of the pocket (most common in shirt pockets), then if there is muktzeh in the pocket, the whole garment can be a basiss l’davar ha’asur (see Shemirat Shabbat K’hilchata 20:74) . Some jackets have such pockets.
There are other reasons for a garment to not be a basiss. A basic requirement of the status is that the muktzeh was placed on the potential basiss with the intention that it stay there on Shabbat (Shulchan Aruch, OC 309:4). In regard to garments regularly worn on Shabbat, the assumption (without a need for cognitive thought) is that one did not intend for the muktzeh to remain there until Shabbat. In that case, even the pocket itself would not be muktzeh. Even if the suit was meant to be used primarily during the week, it still would not seem to be a basiss. Most people do not purposely leave $20 bills in their pocket, irrespective of the laws of muktzeh. Assuming you meant to take it out of the pocket either a few minutes later or at most at the end of the day, there is no issue of basiss whatsoever.
There is a machloket about whether one who is wearing a non-basiss garment with muktzeh in it, has to remove the muktzeh as soon as possible (see Mishna Berura 310:29). (If it were a basiss, this would not suffice, as a basiss when Shabbat starts remains such even after the muktzeh source is removed.) We usually try to remove the object at the first opportunity (Shemirat Shabbat K’hilchata 20:75). However, there are several legitimate excuses to delay doing so (see ibid.). Included in this category are: when a loss will likely occur to the muktzeh object (i.e., someone will take the money); it is difficult to do so without removing the clothing (when the pocket is not a basiss, it is easier to remove the money by grabbing the pocket and shaking out the money); when it will be embarrassing for the muktzeh to fall out or it will be an eyesore. Therefore, it is likely that you could have waited until you got home or to a private place before emptying out the $20 bill.