Can I use a port to run a OLED and read a temperature sensor?

I have a project that uses a OLED 16 x 2 Newhaven OLED, a piezo switch and I would like to add a TMP36 temperature sensor.
The problem is these are remote from the Sanguino and the cable that connects them does not have a spare wire for the temp sense output.

Can I just hook it up to a wire shared with the OLED and read it when the OLED is not been writen to?
If so which is the best pin to use? RS, E or one of the 4 data pins?

Or perhaps its better to use the piezo switch output connected up to the OLED pin's and have the TMP36 on its own pin.

Can I just hook it up to a wire shared with the OLED and read it when the OLED is not been writen to?

It will depend on the devices. In the case where the lcd for example requires strobing, you can do that - assuming that the temp sensor is OK with sending random data to it.