If my data is stationary at the 5% level but non-stationary at the 1% level, should I difference it again (I have already differenced it once) to make it appropriate for ARMA modelling?
What test are you using?
Augmented Dickey-Fuller test
The null of the ADF is that the series is nonstationary. So your evidence is that the data is stationary, but the evidence is not extremely strong. Probably don't difference again.