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Answer: The three requirements (each required by Torah law) of permitted borer (selection), when separating two things that are "mixed together" (ta’arovet) are: 1. One takes what he wants from what he does not want; 2. He does not use a special separating utensil; 3. The separated item will be used "immediately" (l’altar) (Shulchan Aruch, Orach Chayim 319:1-2). You ask about whether one who does not plan to wear the hat for quite a while but is leaving the house now is considered l’altar. At first glance, if you will not use the hat anytime soon, it is not l’altar and is therefore forbidden (how long is a good question – see Rama ad loc.). Possible arguments that it is still l’altar are: A. Since you do not have an opportunity to do it later, the last possible time might be I’altar; B. Maybe leaving the house "properly equipped" is considered a use as you leave. Are these arguments correct?
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There are, however, other reasons that might permit taking the hat, in certain cases. While the gemara talks about separating good and bad (pesolet), Rishonim apply it also to separating two things that are both good (but one is wanted now), if the two items are of different minim (types) (Terumat Hadeshen I;57; Rama, Orach Chayim 319:3). Two different species of fish are two minim (ibid.); one species of fish but one is cooked and one is fried are two minim (Chayei Adam II, 16:5; Mishna Berura 319:15); if the difference between the pieces is size, it is one min (ibid.).
If the contents of the box are all interchangeably usable hats (women are often more discerning), they are likely to be one min and there will be no borer. Poskim say that if one wants to choose a specific book or article of clothing, borer does apply (Shemirat Shabbat K’hilchata 3:68-69). It is unclear where to draw the line on the question of how different the hats are from each other and how thought out your preference of one is (e.g., color match, quality, appropriate for Shabbat). If you cannot determine there is one min, if you are willing to take the first one that comes to hand without choosing (and without repeating until you are satisfied), it is not borer (see ibid. 78).
While we assume there can be a ta’arovet of relatively large objects (see Terumat Hadeshen ibid.), borer is less likely with large objects (see Shevitat Hashabbat, Borer 11). It is possible, then, that if the hats are arranged in an orderly manner, there is no ta’arovet (see Orchot Shabbat 3:23).
There are two possible ways to solve an existing borer problem. According to Rav Moshe Feinstein (Igrot Moshe, OC IV 74), one can undo the ta’arovet by randomly throwing its contents in a manner that the items become clearly distinct, and then there are no requirements. Not all agree to this "divide and conquer" approach (see Orchot Shabbat III:119).
Another suggestion is to "create" an immediate use, by wearing the hat you choose for a short time. It can be a contrived use (see Shemirat Shabbat K’hilchata 3:69), like the Be’ur Halacha’s (to 319:4) idea for one removing bones from meat to lick them before discarding them. A prominent rabbi told me that you would not need to enjoy wearing the hat at all, but there might be some bar to reach (see Piskei Teshuvot 319:12).
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