logistic model effect range

I have a logistic regression model where one of the variables (which is
continuous) has a effect range that crosses zero. The model was created
using R, from which I can produce table which provides predictor variable
details such as low, high,range, effect, S.E., lower 0.95 and
upper 0.95 (for both the variable and its odds ratio).

If the range between the lower and upper 0.95 crosses zero, (or for the
odds ratio crosses 1) this would indicate to me I'm seeing a non-linear
effect, such as a preference for moderate values, etc. BTW, I've noticed
that in cases where this occurs, the P value for that predictor variable
is always very large (not significant).

Can someone confirm that my understanding is correct on this? If this is
the case, should that predictor variable be removed from the model? Are
there methods to transform the variable so that it can still be used?

Thanks

Trevor Wiens
.