Adjust your thinking.
You've noted that you're trying to make some trim pieces. I can't, for the life of me, envision what you need 1x1/8" trim for, but I'll assume you need it and that's the perfect size for what you're after. (Oh! You're making doll-houses or other scale models! Of course.)
I'd suggest that you rip your wood in 1/4" wide strips, ...
Joint the board in sections and iterate through Left, middle, right side until reasonable jointed
but there is a risk that each section is not parallel.
Check often, adjust as necessary.
This is basically similar to the problem of how you face anything that is wider than the plane (a somewhat-common occurrence LOL). Because shavings can be on the ...