By placing the outside pins in an irregular line, your finished piece will not be straight across. You will use the outside pins to create your finished edge so it can be exactly as you want.
The next photo shows two things: you may warp the loom with different types of thread (see the heavier, darker thread and notice it is split; and the placement of the weft material in an arc. Leaving the weft material in an arc will allow it to be "beat" into place without pulling in the side warp giving a more even weaving.
When the weft material is in place, use a comb or plastic fork to "beat" it into place. You can beat it as tight or loose as preferred.
By weaving a craft stick in your weave and standing it on edge, you will create a shed or opening that allows you to move your weft material across the weaving. This will help if you have a heavy yarn or weft material that will not pull through the warp yarns easily. Otherwise you may use the weaving needle to go over and under the warp yarns.
The following photo shows a mostly finished piece. Not the slit on the right, the uneven botton edge, the curves in the weft material, the variation of the widths of the weft materials, and the changes of weft material. Experiment with laying in a heavier weft and filling around with a lighter yarn.
The alternate weaving option is to use random fabric weaving. Strips of fabric are laid across a piece of fusible interfacing. By lifting every other strip and laying a strip across the warp, the fabric is interwoven. The next row will be accomplished by lifting alternating strips. Once the piece is woven, press with the iron to adhere it to the interfacing and remove the pins.
Curves may be cut and interwoven. The trick is to cut the curved strips side by side and keep them in order! You will then put them back strip by strip in the original order.
In either method, experiment with different weaves. The basic weave is over one, under one, reversing in the opposite direction. A basket weave is over two, under two, reserving in the opposite direction. Try over one, under two. Once you have finished weaving, remove the pins. Integrate the piece into a larger quilted piece or use separately. Add texture to your weaving, by incorporating beads, wood, torn strips of fabric or paper. Dare to experiment!