I had the same problem with mine and someone mentioned to thread it a little differently which also keeps the thread from jumping out of the arm that goes and down. I use my D1 for piecing all the time too. The way to thread the machine when you come to the arm is run the thread from the left side instead of the right side so it crosses over itself when you go down to the needle. This has helped me in a number of ways. Also the machine might need to have its upper tension adjusted too. I find if you start sewing with the needle up instead of down the thread doesn't seem to pull out of the needle on its way down. I used to always sew with the needle down and it would always pull out if I didn't leave a long enough tail behind. I try to save thread so instead of cutting between pieces, I'd just pull the last piece a little out of the way of the needle and start my next piece so I'd have a long train of pieces together when I was done sewing. Then I could take the whole train of pieces to the ironing board and press them and finally cut them apart.
If you use the automatic cutter between pieces, it will stop with the needle up. I find my machine will leave just enough tail so it doesn't jump out of the needle for me. Of course right now my machine is at the shop for its yearly cleaning and adjustments. I'm looking at their EPIC machine with my D1 as a trade-in. I'd hate to part with her as she's been a great machine for me for 16 years but with a trade-in of 4000 you can't beat that.
Good luck getting to know your machine as its a great machine in my opinion.