There is no trick, it just won't work.
Synthetic corks are popular as replacements to cork not only because they are cheaper, but more effective at preserving wine as they don't dry out, and they expand more in the neck keeping a tighter seal.
This makes it more difficult, if not impossible, to get them back in.
The simple and easy solution is to buy re-usable bottle stoppers.
There are many different types, I prefer the ones which have a lever or button you push down to expand the stopper as they are best at preventing spills and leaks.
There are vacuum sealers as well, however IMHO
they're gimmicks and don't improve the storage of wine.
cooking.stackexchange.com/users/19707/GdD
One method that has worked for me if you desperately need to use that cork again is to shave the end with a knife to create a taper so that it will slide into the opening of the bottle and then with the pressure of your hand you can squeeze it down in.
I always keep a set of reusable rubber corks around though for this very problem.
They're cheap, come in sets of 2-4 and will fit pretty much any bottle.
cooking.stackexchange.com/users/14601/Brendan
Why not just use a old "Real Cork" cork...
You should still have one around the house...
I save my corks from whiskey bottles, as the have a nice top, and almost always fit into the neck of any wine bottle.
cooking.stackexchange.com/users/10412/Adesso
I can sometimes get them back in by flipping them over and using the corkscrew end.
But if you can, you're better off using one of those rubber stoppers with a vacuum pump (e.g. "vacu-vin").
It will remove a lot of the air, preventing the wine from oxidizing as much.
cooking.stackexchange.com/users/5263/Steve
I just tried it and got it back in.
Put the cork in at a 30-45 degree angle and keep applying pressure.
This works best if you can sit down with your feet under you butt and keep the bottle tight between your thighs.
Apply pressure and take a butter knife and keep working the edges in.
Once in just put your weight on the cork and ta da!
cooking.stackexchange.com/users/43972/Destiny Autumn Bear
Saw a bit of diameter off the cork with a bread knife and happily plug it back in (it may not seal it completely but it'll do overnight)
cooking.stackexchange.com/users/40794/Rehana
I shaved the plastic rind off with a scissors, squeezed it in as far as I could with my hands, then put the hard plastic bottom of a bottle of ibuprofen on top of the cork and pressed down.
I imagine any kind of grippable, flat and hard thing would work.
The ibuprofen bottle made it much easier to apply force once the cork was in a workable position.
cooking.stackexchange.com/users/64788/Meredith
The past couple times I've bought wine stopped with a  synthetic cork , I've had a very difficult time reinserting the cork after opening the bottle. It seems the cork expands after leaving the bottle, and it's made of such a rigid material that sometimes I can't squeeze it back in. (And no, inserting the back end doesn't work, as it sometimes does with real cork.)  Is there a trick to getting an expanded synthetic cork back in a wine bottle?
How to get a synthetic cork back in a wine bottle?
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cooking.stackexchange.com/users/7107/samthebrand
You can sometimes force them in at an angle or upside down, or shave off enough for them to fit. Alternatively, use a cork from a whiskey bottle or a re-usable bottle stopper. Synthetic corks are popular alternatives to real cork as they are cheaper, don't dry out and expand to make a better seal. You can reuse stoppers from other bottles or buy reusable ones. You can also try trimming the cork so it fits back in the bottle.
Synthetic corks are impossible get back in. Buy re-usable stoppers or reuse stoppers from other bottles. Try trimming off some of the cork so it fits in. 