Thanks for your suggestions. Just a bit of background to try and answer specific questions.
Vaguely what I do is test pharmaceuticals (vaccines, chemo drug etc) to make sure they don't contain contaminants and are safe before they are released for use. So, it's very important that we don't introduce any contamination.
The general procedure is into secure unit, clothes off, over barrier into shower 1. Full wash body and hair, dress in appropriate scrubs (none of your own clothes allowed), into level 1. If staying in level 1 just need to add hairnet, gloves, facemask.
If level 2 or 3 then another wet shower or a dry shower (blasts you with air to blow contaminants off you) with full clothing changes in between including respirators, bio suits etc if needed. If you leave (which I need to after I've done my bit of work) you have to repeat when you go back in (I go back in in the afternoon!). I'm glad it's only short term for me and the showers are excellent 
I leave my hair down or loosely plait it. A cap would not be possible, I generally use spray conditioner (aussie - if this is rubbish let me know!) after my first shower and at home.
It's harder to get other shampoo/conditioner further into the unit as it needs to be decontaminated before it goes it too!
I will get my hair cut shorter, last time I did this (need to do it every couple of years, other times I work in a normal lab) I had 11 inches chopped off 
Hair dries quite quickly, it's usually at least 26 degrees in the labs
I just blot it dry and leave it down as long as possible before plaiting it.
I'll investigate the possibility of just using conditioner and/or having my own stuff transferred in.
Funnily enough we have several long haired men who work in the unit though my boss is bald so I always joke that the water just wooshes off him!