If there's a magazine that you've always wanted to write for, sometimes offering the great article as your first piece might not be a good start. Instead, start off small. Get to know the editor. If a magazine asks for letters, or news items, try sending off a few of those first.
Send in useful smaller snippets and the editor will get to know your name. Then, when you offer something more substantial, the editor may look at it more favourably.
The news section in Writers' Forum magazine openly asks readers to send in news stories. Each month, the editor selects 'the best' and the writer receives a year's subscription. I've noticed how it is the writers who've sent stories in over a couple of months (so that's regularly, most months) who are rewarded with the annual subscription. And some of those names have later had full length articles published within the magazine.
One of the travel magazines has a section where readers can send in tips and advice. Again, it's common to see some 'regular' names crop up, but it's also noticeable to see some of those names appearing as the writers of longer travel articles a few months later.
My first published pieces in Country Walking magazine were for some of their smaller, filler slots. I'd sent a couple of reader letters in, and I'd had a few fillers published on their filler pages (news, humorous stories, photos of great views, etc) before I approached them about their walking routes section. Because I was a name they recognised, they agreed that I could have a go at writing for this section. Nine years later I'm still writing for the magazine, and have helped them out with a couple of last minute deadline pieces, too.
Sometimes it can be worth thinking about what you want to achieve with your writing. Consider it as another relationship, and begin it as you would any other relationship. (Don't go for a seven course meal to start off with - meet up for a drink first!) Start off small, and see if you like each other. Who knows, it could be the start of a long and fruitful relationship.