The method I have used when needing to use a piercing saw at an angle is to twist the (soft) ends of the blade. As long as your blades are not rusty (and have been properly hardened – not at the ends) this works well.
Another hint which might help is using a piercing saw for parting on a lathe with a very narrow waste band: Ensure that the blade is fitted the correct way, teeth pointing towards the handle, and use the saw below the work. Either spin the work fairly quickly and move the blade slowly back & forth, easing pressure on the return stroke, or rotate slowly and saw quicker. Don't leave the saw stationary in contact or the teeth risk clogging and jamming. And don't try this on thick stock, of course.
Traditionally saws were available with a working depth of 6 inches or more.
Cheers, Tim