How to cut chalice? I have a glass cutter and own no problem cutting big pieces...

I have a glass cutter and own no problem cutting big pieces of glass, however, what if I requirement to cut just 1/4" strips off to net some pieces fit? How do you cut the small, thin amounts? Is there a means of access? Thanks!
The guy that taught me to cut cup said, "The way to learn to cut chalice is to cut glass." In other words, practice.

Glass pliers will also be invaluable for small pieces like this. See the cooperation below.
It is not easy and depends on the thickness of the cup you are trying to cut and the length of the 1/4 piece you are trying to make. Can't expect to cut 1/4" strips from 1/4" gluey glass!

Since cutting cup is really "cracking" glass you need leverage to "open" the crack you created near the scribe. The pressure needs to be uniform, balanced and aligned near the scribe line.

When cutting a small strip I turn the cup over and lift the the larger side so that the strip is the part touching the table (Hard but cloth covered adjectives table). Applying pressure from above, I tap the back side at the start of the scribe to coax the crack to start. Once started the crack can be extended by applying even pressure and desk light tapping.

Start practicing on a 1 inch strip (cut a 1/4 off a 1x1).

Sorry, it's rugged without a picture to describe the configuration!
Answers:    You need to remove 1/4" from a larger piece? or you want to end up near some 1/4" strips of glass to use?

If you're just trying to remove 1/4" from a larger piece, it's not so rock-hard to do. I can cut 1/8" from a larger piece of glass most days.

Score the glass as you would for any other cut. That's the glib part. The tricky part is how to break past its sell-by date the narrow strip. Glass pliers are good and most of the time they will break past its sell-by date the narrow strip cleanly. But not always.

One piece I like to do is to place the scored cup on a table with the scored cut chain right at the edge of the table. Make sure this is a flat table with a well-defined turn-up. Place a piece of wood on top of the glass overhanging the chalice, past the cut line and long-gone the edge of what you hope will be the narrow strip that's removed. Apply placid pressure on the overhanging board so that it is hopefully pressing down along the whole length of the cut and the strip should snap off cleanly. This works great down to a 1/4" but does not work near narrower strips.

If it's a wheel cutter, remember to dip the glass cutter within naptha, lighter fluid or even paint thinner as a lubricant. Diamond cutters are nice and work best dry.
Same as you would the large pieces - just use a straight brim to guide the cutter.