It depends on so many factors...
If hedging out a blue chip and for some reason you couldn't hedge with that same product at some point, you would get creative. One example might be hedging via a basket of highly correlated stocks until you can hedge properly.
If hedging out a micro cap stock with low OI it will be done via phone. If a client inquires about it, then you will put the feelers out with other MMs and see if any clients are holding this particular stock. In most cases someone will find someone who has some and you add on your spread to the bid/ask.
To answer a more broad Q from the OP...Every MM will vary depending on in house risk management practices but from past exp. - using FX as an example, you will have limits on the size of a position you can build up directionaly for each ccy. For G10 your limits will be much higher than exotics for example, and flow will usually net off to some extent, but if your book starts getting heavy on one side, say you are Long USD, you will start posting really attractive Offers on USD in order to attract flow looking to buy USD from you. If you breach limits you will need to hedge out risk asap and as your book moves against you you will need to start hedging out as well to mitigate losses.