As someone who covers her hair for religious reasons, occasionally with a wig, and lives in a community of other women who do so, I have to echo Kaio. With a good wig (high-quality, unprocessed human hair, monofilament top, made by someone who knows what they’re doing), it shouldn’t be at all obvious except perhaps to people from wig-wearing communities. (Trust black women and Orthodox Jewish women if they say someone’s wearing a wig. We know.) If this is something you’re really committed to doing, spend money, or everybody will make fun of you behind your back for your bad toupee. My wig cost $2400, but a short one should be much, much cheaper - $400-500 or so.
That said, what texture is your hair? I have long, thick, wavy-curly hair, and I have to keep it slightly shorter than I might otherwise like in order to get it up under my Fake Hair without it making the whole thing sit up too high on the top of my head. This still means that my hair is longer than the bottoms of my shoulder blades, but my favorite mid-back length is a bit longer than I can stuff up under a wig. Your hair is probably thinner than mine, so you might not have this issue.
There are various ways of putting long hair under a wig, although I’m not sure how well all of them work under short wigs as I’ve never tried it. I currently switch off between pinning up my hair in 10-15 sections with bobby pins, getting it as flat as possible against my head (more comfortable, bad for the netting of the Fake Hair, and takes ten minutes), and using a wig cap (fast, easy, leads to a slight headache from head-squeezing, and will probably be hot in the summer months). Many wigs also have a small pouch in the back, designed to hold your hair; my wig has this, but my braid kept falling out no matter what I did. Putting my hair up in two small buns (get your mind out of the gutter) at the base of my neck works for me, but wouldn’t be hidden under a short wig, plus I really don’t like the sensation of the buns pressing into the nape of my neck whenever I look up at all.