Since your job hunting is still months away, start now by gradually improving the way you dress. You may still garner a little suspicion but if you go slow, nobody should notice.
- Start with nicer shirts. If you wear T-shirts, start wearing polo's. If you're wearing polo's or other business casual wear, start with a dress shirt.
- After a few weeks wearing dress shirts, start wearing dress slacks for a few weeks.
- If you wear very casual shoes, do a slight upgrade on those. Also wear nicer socks.
- If you are planning on changing your hair at all (tidying up, clean-shaven, etc.) plan that somewhere in there too.
The idea here is to inch toward wearing dressy clothes without actually being dressy. Even in casual environments, it's not unheard of to wear everything but a jacket and tie. This allows you to wear part of your suit to work and since you've been dressing like that for a couple months, nobody will be the wiser.
Yes, someone may well notice you're starting to change your clothes style. You can pass it off as dating someone or (if you have one) your significant other has been pushing you to dress nicer.
You may even have someone ask you directly if you're looking for work. You can say that you aren't, guilt-free. Then you continue wearing the nicer clothes.
The key is to avoid sudden changes.
When the time comes to start with the interviews, you being out for personal appointments won't be as suspicious when they look at your clothes. You can take a half day here and there for the interviews with less suspicion.