What you want is a harmonic exciter. Some are called tube, preamp or console emulators, but they all do essentially the same thing: generate gentle harmonic distortion.
There are many to choose from, ranging from
free to
cheap to
fairly expensive. Some are
one-trick ponies, some are quite versatile.
Here's one that's both inexpensive and fairly versatile.
My personal favorite is rather pricey, and that's the standalone exciter module bundled with Ozone Advanced. Second-favorite (and second in price) would be FabFilter's Saturn. Those two account for 95% of my distortion-lite needs.
P.S. Don't waste your money on the BBE Sonic Crapifier.