Are you at least mixing at a proper dub stage?
Ok, so if you only have six days...
Yes you need to focus on parts at a time.
Having six days you are then likely to have a maximum of five actual mix days.
Last day for screening and small fixes.
With such limited time you will pretty much have to go straight for final mix.
The advice below is all given according to this limited schedule, a more realistic mix schedule would be handled differently.
Make sure that the film is "gridded" with markers or dummy events so that you immediately can identify the length of a scene and find a cut without searching/scrolling to find it. Try to premix before hand as much as possible, you have more important things to do than choose reverbs, arrange basic routing and sends during the mix, so having the foundation set is helpful. Mix in the box! Don't use a hardware audio mixing console as that will make it hard/impossible to do any extra fixes and or versions after leaving the dub stage.
I'd probably use about 1.5 to two days with focus on dialog, make sure levels, EQ, noise reduction and reverb is appropriate and that no major issues are left. Then spend two+ days for the full mix, avoid messing with details and focus on the overall.
Do a time budget. For a 90 min film you have to complete 45 mins of dialog a day, approx 5mins/hr (depending on how long your days are). Do not overspend your time. If you aren't satisfied, just drop a marker and move on. IF you have spare time somewhere, go back and revisit. 5 mins per hour might sound ok for someone that has only dealt with smallish projects. But you can only play any single scene a maximum of twelve times if you never would stop and never would have to spend time to adjust stuff. In reality you have to of course so if you have seen the scene six times in a day, then you are probably close to the limit.
Depending on how the mix stage is set up you will be more or less efficient as you are just one person. A controller based setup optimised for a single mixer is likely to be reasonably fast to work with, a multimixer dub stage will likely bring you down if you really need to work alone.
You have a director. Obvious right? He or she is your boss, it's their vision that you need to fulfil not your own. Make sure you communicate and that you plan enough time to play through what has been done and that you explain briefly what is done and what isn't, make sure you have time for discussions as well. Make ure they understand YOUR schedule and how you plan to use your time.
Either have the director come in each day for a play through and to write notes, or have them in the room from around day three. You don't want any major last minute surprises!
Make sure that the director understands how much time you have available to mix the film.
Be 110% sure that all sound elements are included with the project/session! Open the project without any connected fx hard drives or servers to be absolutely sure.
Make sure that the dubstage have the plugins you need! You don't want to spend time installing and uninstalling stuff.
Make sure you session is arranged in a way that you can deliver material according to the delivery requirements with as little extra work as possible.
Only drink coffee while you watch a completed segment. :)