Try the distortion 2 plugin with a gentle setting, that will warm things up a little.
Also try the auto filter plugin, that has some inbuilt saturation settings you can play around with.
I'd recommend you to also try Logic's compressor, it has some clipping functionality that you can employ, you can also switch on the inbuilt limiter section, bring down the threshold to gently clip the signal, that will get a warm sound going on