If not, can I go to another life insurance company and get a term life insurance in addition to the whole life that I already got from work?
There's no limit to the number of life insurance policies you can buy. And when it's through work, you don't get to choose the type of plans available, your employer does that. So you can't call and switch that employer benefit life insurance coverage, but you can go to a local life insurance agent, and get quotes to buy your own life insurance separate from your work plan.
The insurance company will ask on the application about other life insurance policies you have and how much coverage, to make sure you aren't over-insured. But, most insurers will allow up to 10 to 20 times your annual income without any problem.
Here's how to learn more about term life insurance and compare free term life insurance quotes online from several leading insurers.