Another example:
Your home field doesn't have an instrument approach procedure but an adjacent airport has an ILS, LPV, etc. You file to the better-equipped airport, fly the ILS, LPV etc. approach at that airport until you break out of the clouds, then request a contact approach over to your home aerodrome. Note: Best to notify ATC of your intentions prior to the approach. 1 mile visibility and clear of clouds is all you need.
Word to the wise: This procedure can be very useful but can also be very dangerous. You're basically scud running below the clouds, at a low altitude, with low visibility which creates the possibility of you hitting something on the way over to your home field. Also, the weather can close in on you at a moments notice and your IFR again with nothing more than a contact approach clearance. You are now violating regs. Not good. If you know the terrain extremely well, know where every obstacle is located and are confident the weather is not rapidly changing, contact approaches can be very useful, especially for those of us not based at a field with a SIAP.