Exterior Door Width is Shorter than Jamb

I have an exterior door with a large gap to the strike-side jamb. When closed I can see a 1/4" inch gap from top to bottom. Fortunately the dead bolt works but the latch bold is just hanging in there. It’s as basement door and I am need to fix it before casing it.

I was thinking of adding 1/4" strip on strike plate side to make jamb smaller but not sure if that right way to do it. Try to search on line but was not able to find answer for my specific issue.

Also, thinking about weather strip that I may need to take out and replace it back on top of new 1/4" strip and what about outer side. Do I need to add the strip on outer side too?