While breaking the chain, I pushed the pin too far and pushed it out completely. Now the chain is too short. How do I re-insert this pin, or should I look for other options.
What type of chain - Shimano have chain connector pins for exactly this task for many of their chains. Note the pin must exactly match the chain. If you have a length of the same chain (I always keep the left overs when I put ion a new chain) break the chain again and remake the chain with the leftovers.
I have (in desperation - bike shops 100km away, no spare chain parts), used a vice and pointy grips to do press the pin in - the repair was dodgy but got us on the road.
Alternately, use a quick link or a new chain.
I always carry the cutoff from a new chain (or a few links of it) in my kit. Take the chain apart at the next link down from where you lost the pin (being more careful this time) and then take two half-links from the cutoff and install them.
And, while you have that cutoff piece handy, do a trial to see how many turns of the crank it it is to take the pin almost all the way out, write down than number, and store it with the tool.
Or just use quick links.
I just tackled one of these myself, so I thought I'd share my alternative experience.
I am short on tools, so in absence of a decent pair of pliers/hammer, I put the pin inside the inner chain to hold it in place and positioned it on top of one of the outer chain's legs (resting on the ground or an elevated surface). Note that this might require you to reposition/turn the outer chain's legs so that you can access the the hole from straight up. Then take a flat, metal surface (I used a dull bread knife) and place it on top of the pin positioned over the outer chain hole. The idea is just to apply a lot of pressure at as straight an angle as possible. Since it's a flat area you are applying force on, you can really put some muscle into it. For me, the pin slid in a bit within a few tries of repositioning.
From there on, as soon as it's a bit in, it's the traditional stuff.