Yes, I would start with the preamp. I am not suggesting that you spend your entire $6k budget on that component, unless it makes sense. It might make sense if you expect to have more budget in the foreseeable future, if not, you may have to scale your purchase. My point was that you should make the changes one step at a time, so that you can get a feel for the impacts of each component as you add it to the system. Keep in mind that the first upscale preamp you buy may not be your last, and you may have to try a few to get one that best fits your system. I am on my fifth, but pretty satisfied. Best to go used for the sake of economy.
A lot of recommendations for tube preamps, and I don’t disagree with that path. However, make sure it fits your lifestyle. My gear is on all the time and sometimes plays all day. For that reason, I have not ventured into tubes, though I did have a Schiit Freya that gave me the option use them.
Be sure to allow some budget for decent interconnects, they will make a difference as you get into better gear. Acoustic Zen and the newcomer Veritas, are both reasonable and decent.

