K1000 has the same, standard film rewind button on the bottom of the camera, and the K1000 manual has a page (#23) 'MULTIPLE EXPOSURE' telling you how to do it:
Quote: "For deliberate multiple exposures, make the first exposure in the normal way. Then tighten the film by turning the rewind knob 'A', and keep hold of the rewind knob. Depress the film rewind release button 'B' and cock the rapid-wind lever. This cocks the shutter without advancing the film. Finally, release the shutter to make the second exposure. Then make one blank exposure, before taking the next picture, to avoid overlapping as registration may not be exact."
It seems that the issue here is that even when the film-rewind de-clutch has been engaged, there's still enough friction remaining on the film advance when the shutter is re-armed that if you don't hold the film in place with the rewind knob, it may partially advance, and your second exposure won't line up with the first.
As far as I know, all the manual film-advance cameras work the same way. Looking at the instructions for the LX, it is similar in concept, but a bit better because the exposure counter & advance is more robust & precise so you can relatively easily rewind backwards multiple frames by watching the counter -- the counter will count down backwards with high precision as you rewind film (they claim 0.2mm tolerance!). This is detailed on page 32 of the LX manual.