There's no need for all this hammer and screwdriver stuff. It's a little fussy to get the pin restarted and you'll want to use your extra hand (the trusty bench vise - EVERYone doing more than basics should have one) to hold the chain itself while you work the pin pusher back and forth to reset where you're centering the push to that you can work it in as straight as possible. If it's out by a few degrees then it'll straighten itself as mentioned but if it's out by enough to clearly see you should reset the position of the pushing pin to get it more straight. A pin starting in crooked will deform the plate more then neccessary and then it WILL be weak.

Once it the link will be tight. But our chain tools all have a set of fingers further towards the pushing pin for this. Just flip the chain tool to push from the other side or catch the first side a small amount before the pin is all the way home. Put the chain onto this second set of fingers so they are inside the plates and use the pushing pin to move the pin just a 1/2 a hair more. The link will now be as flexible as those around it. If not then push it the other 1/2 a hair more.

Is it a good idea to push out a pin and then back? No. You WILL do some damage to the plates. After all the chain is made in the factory by swaging the ends to make them larger AFTER the pins are put in place. So mashing the larger ends through is not going to help things. However I've done it and others have done it and Shimano manages to swage them only just enough that they are meant to be done this way. So as always with stuff on the edge like this YMMV.

I suppose that if I was some ToD or Tri athelete with legs the size of building columns then I'd never do this and would just buy a new chain rather than take the chance. But I'm just a whimpy 54 year old commuter that is happy to see the odd sprint to 25 mph on a good day downhill with a tailwind. So I've never had any issues with my pushed back pins on the odd time I've used them.

But PLEASE put those hammers, steel plates and screwdrivers away. The chain tool will do the whole job without the risk of any impact forcing things that don't want to go and possibly deforming the parts so that they need to be forced that much more. The only thing that would be handy is to hold the handle of the chain tool in the vise and then use one hand to screw in the pusher while the other uses a pair of needle nose pliers to hold the pin as straight as you can eyeball.