Did you try some plots? Show us. And, partial correlations correspond to correlations of residuals, see https://en.wikipedia.org/wiki/Partial_correlation. So, the corresponding plots woud be plots of residuals, which very well could be helpful.
More details. With variables $x,y,z$ we want to "partial out" $z$. Then estimate the regressions $$ x=\alpha_0+\alpha_1 z + \epsilon_x \\ y=\beta_0 +\beta_1 z + \epsilon_y $$ with residuals $\hat{x}=x-\hat{\alpha_0} -\hat{\alpha_1} z$ and $ \hat{y}=y-\hat{\beta_0} -\hat{\beta_1} z$. Then the partial correlation is $$ \text{cor}(\hat{x}, \hat{y} ) $$ and you can make a plot of this residuals, which will show any nonlinearities. All of this still makes sense if the linear regressions above is replaced by nonlinear regressions.