You sound like a caring an patient momma. I never used time-outs because to me they are "pretend" consequences that are too lenient. Some kids hate time outs, but difficult kids have as long as they want to get tired of them with no real consequence and they enable tantrums. You know your own child. You call it "she gets more upset" in the time-out. I would consider it her having a tantrum because you are disciplining her, which is a NO WAY in my house and the tantrum would be disciplined on it's own. That goes for any form of discipline If the child throws a fit about it or pouts and uses disrespect, NEW CONSEQUENCE! And I also never ignored or let tantrums play out in a room. If you stay engaged and discipline bad behavior immediately after a clear warning, then the child will start heeding warnings much faster.
At six your daughter is mature and developed enough to understand right behavior and act out of mutual respect for you if you are EFFECTIVE in disciplining this very bad behavior. She is also old enough to start acting out badly if you aren't effective enough. In my opinion books like How to Talk so Kids Will Listen and Love and Logic are excellent books, but not for nipping very bad behavior. They're great for the long haul if you can hang in there or for after you nip the bad behavior as a guide for every day niceness to avoid conflict. But I would be tougher up front so she stops the behavior and then you can revert back to those nicer options, which are really just basic nice respectful ways to handle your child, which most good parents use...until the kids is off the rails...and in that case, the advice doesn't help much.
I've heard friends of mine say it's normal for kids to say "I hate you" and they just say "Well I love you" and ignore the behavior or tell the child it's not nice...this blows my mind. I can't fathom what my parents would have done if I was ever that disrespectful. Now maybe as a teenager in a shouting match, but not anywhere in the league of 10 and under. My kids would NEVER say that to me. And I would never say it to them. Absolutely not allowed in our house. They have said it on occasion jokingly, and then I say in a very serious tone, "Actually that's not funny, we don't even joke like that." And that's all it took for them to never try it in a serious way. As a tiny child, absolutely nipping is best. As they get older, you can relax more into a groove and joke around and let them express more loosely, but 6? NOPE. These are formative years and you can't let that develop or you will have a terribly disrespectful tween, teen and beyond, which is great that you are reaching out.
My oldest daughter is 7 1/2 and very respectful, does not throw fits, does not speak meanly to me or anyone else. But that's because I handled her very first attempts firmly as young as 3 years old. One calm warning and one FIRM consequence the second time. Now that she's old enough for logic and empathy, it never comes to that, we can talk things through and she wants to be nice because it's what we have modeled and it' all that is allowed. If she starts to snark, just one, "Hey, is that how we talk to people?" and she tones it down. I also be sure not to be rude and snarky to her.
The book I used is "Back to Basics Discipline" by Janet Campbell Matson and it worked beautifully on my three very different children-even my extremely difficult third. I have a tougher book too that I used if you PM me. You can remain positive and kind and calm and non-angry if you have an efficient system, and the book helps with that. I never discipline in anger, and because of that, my kids heed quiet warnings very well.
Some of my daughter's peers are acting the way you describe, and I've heard their moms comparing notes on how their daughters are getting worse..it's a normal phase at this age, and it's totally up to you how bad this gets. I would tolerate NONE of the disrespect long enough to put it on a chart without a serious consequence. If my daughter pulls a huge pout because I sweetly tell her to take out the trash and clean her room before she turns on the TV, I say "And smile about it!" as a joking way of saying, "watch it, sister." My dad used to do that too so it's a family tradition :)
It's much easier to nip things when they're small then to let things like "I hate you" slide and then try to deal with the bigger outbursts to follow. It's been easier with my younger two because they all know no one is allowed to be mean to momma and they are uncomfortable when they see friends smarting off to their parents. It's all in the foundation years, it's too late once they get a little older-good work addressing!