Proof-reading and editing (or copy editing) are actually different though overlapping activities, arguably requiring opposed skills. A good editor is creative and imaginative; a good proof-reader adheres consistently and strictly to style guidelines.
Assuming you have neither person at hand, here are four tips that might be of use:
Show your work to people you trust and who seem like appropriate people to show. You don't need to rely on a professional editor in the first stage. I say people you trust because they should be people who are honest with you and will be critical because they are trying to help you. I say 'appropriate' because different people will have different perspectives. Some things will be fairly uncontroversial errors ('move' for 'movie' or vice versa, for example). Others will concern the clarity of a piece, it's structure, which is to say that good editing goes beyond proof-reading.
Ideally, you should want to get as many perspectives as possible. I've shown work to many friends and colleagues, some of whom don't speak English as a first language, because sometimes they will read more carefully than those who do. But realistically you don't want to spend too much time quibbling with lots of people over various minor aspects of your writing. So select four to five people if you can and make sure you're willing to return the favour sometime.
Read out loud. Or get one of your friends/colleagues to read out loud to you. Simple errors should become apparent and you will also get a feel for the clarity and rhythm of the piece which a good copy editor would also be concerned with.
Spend time away from the piece. Getting some distance from a piece can be a good way of seeing things and spotting errors that hadn't reached your attention before. Especially if you can't find four to five people to help you with the reading and checking. The idea is to get different eyes on the document, eyes which will spot things that your eyes will have missed. You mention going back to a piece the next day but giving it longer might be better still. It depends on your deadline of course. I've left a piece for a year before, whilst getting on with other things. Your eyes in a month's time will be different (through experience, through taking a break) to your eyes now. For one, they will be less pink with strain.
Accept that you will miss something, that your work could always be better. I'm reluctant to advise against striving for perfection; as you say it's no bad thing, and I think there's even something admirable about it. Attention to detail can make a piece of work very special. Many won't go as far because after a while the returns seem to diminish. There's an element of self-sacrifice as one spends longer and longer on the little things, even to the point of a kind of madness. Go as far as you're able but when you finally declare the thing finished, be ready accept that anything after that was beyond your ability to notice at the time and move on. With practice, you will get better at striving for perfection.
On that final note, this quotation from the poet Paul Valéry might be worth putting up somewhere you can see it:
How can he recognise that his work is finished?
That's a decision he has to take…In fact, the completion of a work is
only ever an abandonment, a halt that can always be regarded as
fortuitous in an evolution that might have been continued.
—from 'La Création artistique' in Bulletin de la société francaise de Philosophie (1928), via Reading Paul Valéry: Universe in Mind, Gifford, P. and Stimpson, B. (1998). Emphasis in the original.
