Yes! You can add an additional discount code at checkout to combine with Fast Bundle discounts. Just make sure discounts are combinable.
Turn on Discount Combination in Fast Bundle
Go to Settings > Bundle Settings > Discount > Discount Combination and choose this option: 'Allow bundles' discount codes to be combined with other product discount codes which are combinable'
Make sure your discount is combinable with Product discounts
Fast Bundle discount codes are product discounts, so make sure your discount code (automatic or otherwise) is combinable with other product discounts in Shopify.
Example
In the image below, a discount code "Test_Discount" which is an order discount has been applied at checkout and is combined with the bundle discount.
Order Discounts
Right now, if you make a Shopify order discount and set it to be combinable with "Product discounts", the order discount may not combine properly with the bundle discount since this feature is new. If the combination didn't work, you can contact Fast Bundle support to check whether it's possible to manually activate the order discount combination issue for your shop or not.
For more information on Discount Combinations, read the following article: