Pennyhal

I have a sewing machine that has an embroidery module. I've been collecting quilting designs that I can use to make a quilt, but I can't figure out just how to hoop the fabric and stabilizer and backing when doing a quilt block.

I have a roll of washaway paper stabilizer that washes away and leaves a flurry of snowflake like fluff like you'd find in the wash after putting a kleenex through the washing machine when washed. The fluff is suppose to just add a little extra batting to the bat that I used for the quilt. Then, I have to add the backing to the quilt block to keep the snowflakes from getting all over in the wash and clog up the drains. However, the hoops for the machine will not hold that thickness, so I had to hoop a piece of washaway stabilizer and them pin the quilt sandwich to the washaway. So it goes like this:

Quilt fabric
batting
leave in stabilizer
quilt backing fabric
washaway stabilizer

This just can't be right! There has to be a better way to stabilize the quilt in a hoop. Can anyone share how they do it?