I just went through this process myself on a different Homelite UT33600a trimmer. First you can't trust all parts websites because they will show the wrong part numbers. I've had good experiences with Jack's small engine website. All you have to do is type in your MFG and model in Google and Jack's will be one of the hits. See below for the link I got. According to Jack's the part number is 309562007 for the spool head. Then I usually search Amazon or eBay for the part. I have a Prime account so shipping is free and returns are no hassle but I did end up paying $4.20 return postage on a $29 gas tank.

Amazon.com: Homelite 309562007 - Original Reel Easy Trimmer Head - Ryobi & Craftsman OEM : Patio, Lawn & Garden
Amazon.com: Homelite 309562007 - Original Reel Easy Trimmer Head - Ryobi & Craftsman OEM : Patio, Lawn & Garden
www.amazon.com