Rules of thumb for partial residual (component + residual) plots as diagnostics for linearity?

Here are the standard R diagnostic plots of a multiple linear regression model that includes an autoregressive term at lag-1 (i.e. AR(1)). I have logged & z-scored my input data. Ben Bolker says here that a scale-location plot is good for determining heteroskedasticity, and a residuals vs. fitted plot is better for determining linearity. So my interpretations of these results are that the multiple regression is pretty linear (residuals vs. fitted plot), and normal (Q-Q plot), essentially homo-skedastic (scale-location), and the outliers aren't too bad (residuals vs. leverage). So far, so good. But when I do a partial residuals (component + residual) plot, the plots for the individual variables show that none of the component variables are linear: The dotted red lines show the least squares fit, and the green loess smoother lines, as I understand it, indicate the real shape of the data. John Fox's book Applied Regression Analysis and Generalized Linear Models, 3rd ed. in Chapter 12 shows some component + residual plots that he says should be data-transformed for not being linear, but his examples don't show the zig-zag pattern I'm seeing in these plots. So these seem worse than the ones he shows, but on the other hand, maybe the diacy.tmin plot is close enough to linear, even though it wiggles around the least squares fit.

My question is: how bad do the components + residuals plots have to be before it's necessary/advisable to transform the data to improve linearity? Are these plots too problematic to leave in the model as-is? And because the first set of diagnostic plots are well-behaved, and presumably show linearity, does that mean I don't have to take the components + residuals plots as seriously?