I don't consciously insert symbolism on the first draft, but I find that it crops up naturally. So after my first draft is written I go back and take a look at it and see where recurring symbols or images pop up. Then I play with them and reinforce them, adding in more where it works.
For instance, in my WIP, I noticed a lot of the scenes and setting involved growing things. I consciously had one character start working in the garden as a symbol of his decision to finally start emerging from his shell. But then looking back I realized that I have another character look into a dilapidated fountain and notice that life is actually coming out of the stagnant rainwater - moss, tadpoles, etc. And when the two characters come together, they end up walking through a park. So I went back and changed things around a little... I took a third character from one setting and had him do the same scene, but at the fountain. I decided to set the garden on fire at the "oh no it's all going to fall apart" moment. Etc.
In one of my short stories, I carried through the symbolism of green eyes. The MC is mourning the loss of his wife, who had green eyes. A woman he meets in the story is wearing green contacts - she is the "false" woman who cannot take his wife's place. And when he punches his brother in the eye, he gets his brother a package of frozen green peas from the freezer - his brother replaces his wife.

None of that was there on the first draft; first I wrote the story, and the only thing that was there initially was the wife's green eyes. But then I needed him to get his brother something for his eye, and I thought, hey! Green peas. And then it clicked with the other woman.
So for me, inserting symbolism is mainly looking for what's already there naturally, coming to an understanding of WHY it's there, and then amping it up.