Hi,
(((Hugs))) to you. I went through a LONG training process with my DS so I know exactly what you are going through. I have several things to suggest to you. I just hope I can get them all down before my lo wakes.
1) To start, your lo is not getting enough night sleep. She should be getting close to 12 hours and is getting closer to 10. The reason she is waking at 5:30 is because she is overtired. Overtired babies tend to wake earlier than they should. I am going to assume that you simply cannot put her down earlier or try to have her sleep later since you've mentioned your work schedule. Ideally you would have her down no later than 7 and have her up no earlier than 6 but we'll see if we can improve her day sleep to make up for it.
2) 6 month old babies can stay up around 2 hours before needing a nap. Typically, most babies prefer a shorter first awake time. I recommend getting your baby down for first nap no later than 7:30 if she is waking at 5:30. The reason her nap is short is because she's been up 3 hours, is likely way overtired by then and isn't rested enough to transition to the next sleep cycle.
3) Tracy cautions against any form of crying it out/controlled crying because she feels it breaks trust between parent and child. Please do not allow her to CIO. Pat/shhh and PUPD are techniques that reassure your baby of your presence while also teaching them to self soothe.
4) Pat/shhh and PUPD work if done correctly and on a consistent basis. It is very important that anyone caring for your child learn how to do this properly. It will be very confusing for your lo if each person puts her down differently. When you say she doesn't take to PUPD, how are you doing it? Babies 6 months and up settle better OUT of your arms. So if she's able to sit/stand in the crib, WAIT for her to do so, then just lay her down again gently and talk reassuringly "it's okay, time for sleep". If she can't sit or stand then PU and calm her, but do not hold her too long. This is one of the biggest mistakes people make. If your lo starts to settle in your arms then she isn't learning how to self-soothe. Please visit the PUPD board if you need further clarification on technique.
5) What temperment is your lo? Touchy, Grumpy, Angel, Textbook, Spirited? Now that she's becoming more aware of her environment she may need a darker room for naps, particularly if she's spirited.
6) It's okay for your lo to cry a bit before falling asleep. My lo always lets out a little waaaaaa when I walk out of the room but then settles within a few minutes. It's their way of shutting out all the excitement of the day. So if your lo is only crying a few minutes before falling asleep, that's okay.
7) In order to get longer naps, you will need to extend them using wake to sleep, pupd and/or pat/shhh. I recommend watching one of her naps and seeing what time she starts to stir. You want to go in 5 minutes before this happens and lay your hand on her back or chest or even on her arms and legs. This will help prevent her jolting and help her transition. When she's better rested this should be easier for her to do as well.
Here is an approx. sample schedule that I hope will work for you.
5:30 wake
7:30-9:00 nap
11:00-12:30/1:00 nap
2:30/3:00-4:00 nap
7:00 bed
I hope this is helpful to you. Let me know if you have further Qs. My connection has been touchy, so I may not respond immediately.