blue chip usually = suit and blouse, at least for the interview - you can always dress down day-to-day if the office is more casual than you expect.
In this weather I'd wear a sleeveless blouse and leave my jacket on, but you could wear a smart shortsleeved blouse with no jacket if it's absolutely roasting, so long as it's no obviously half-a-suit.
Smart dress could work too - 3/4 sleeve wrap dress with a collar might do, but nothing too flowery or frilly.
My immediate thought is it's generally the little things that make the difference - ironed shirt, polished shoes, do your hair properly (large clip always looks better than an elastic if you wear it up/back), make sure nails are clean and tidy if not actually manicured, appropriate jewellery, invisible underwear, good bag (polish that too if it's leather), enough makeup so you don't look haggard but not so much you look like you're going out.
If skirt, closed toe shoes and nude tights (I know, I know), if trousers I'd still wear closed toes.
um, run out of steam a bit here - will post if anything else occurs!