4pm is an odd nap time. I would set sleep times that are good for you. So maybe 9ish am and 2/3 ish pm.... only if she tired though. My first 2 dcs would have an 11am sleep for about 2.5 hours and no more at this age. But they were in bed from 6.30pm.
Maybe 8pm bedtime is just too much for her then......
I've tried to see her sleep cues and in the mornings it works a treat but the same cues are shown at around 4:30 - 5pm each day, I lay her down and she starts screaming which then wakes her up and she gets worse and worse.
We don't always go out but on the days we go out it doesn't seem to make much of a difference to her sleeping so I didn't see the point in trying each day... esp as there isn't anywhere nice to walk around here!! lol
WHy dont you try to stretch her morning nap and then give her tea at 4.30 start the bedtime routine earlier and put her to bed at night time at 6.30 we did this for DS as he was getting overtired and found it made a huge difference if we just started the bedtime routine earlier.
I'm gonna try it when we get home.. I think while we're away I'll stick to the tried and tested thing of putting her in the car for an hour in afternoon for those 2 days... then when we get home, She will sleep in the afternoon or sleep longer in the morning and she will sleep all night.
BTW, while i'm on it - what do you all do if they wake in the night?
having a routine at 9 mo is not an extreme idea but it should be a routine that fits in with your DD and is based on how she is through the day, not a routine lifted from a book. I struggled to get DS1 in to a set routine as a baby. With DS2 I went with the flow a lot more - just did whatever worked to get us through the day and he sort of fell in to his own routine in the end. Good luck