So I saw the hat and it is cute! So I was trying to think of how to do it because I think you can get the gray part done all in one pattern without having to go over it with a trellis pattern on top. So I took some spare yarn and made a chain of a number of stitches (I just randomly made a chain) and then connected it to the first chain to make a round with a slip stitch. Then I began with a chain 3 and did double crochet (one stitch per chain stitch of the original round). But this is how I did it and made the trellis on top:
Round one: Start with 3 dc, treble crochet into the front of the THIRD previous stitch (so count back three stitches and crochet into the chain there) and then SKIP 2 stitches of the chain and treble crochet into the third chain. Then Go back to the previous 2 chain st that were skipped and dc in each one (we are now back to normal double crochet by going right to left) but make sure that you are crocheting behind the treble crochet stitiches so that they stand out on the right side of the work. Skipping the stitch that has the treble crochet in it, dc the next three stitches and then repeat the treble crochet process as written above. You will notice that each treble crochet that goes back or forward will be meeting the next treble crochet and creating "V"s on top. You continue this process until you finish the round, connect with a slip stitch and then chain 3. (you are now about to do Round 2 and you don't turn the work) Dc the next stitch and then skip 2 stitches and treble crochet into the next stitch (which should correspond to where the lower treble stitches make an upside-down V), then go back to the skipped stitches (and staying behind the treble stitch) dc 6 stitches. Count back 3 stitches and treble stitch (in front of the work) to where the lower inverted V is. You should now have your first completed "X" of the trellis pattern laying on top of the underlying dc pattern. Skip 2 stitches and treble stitch into the 3rd stitch and then go back and (beginning in the skipped stitches) dc 6. Continue in this manner around until you finish round 2 and join with a slip stitch. You will continue round 1 and 2 in this fashion until you get the desired length.
I hope I explained this OK. What I wrote above isn't perfect. In general, I think that if you figure out a good ratio (like 3 dc going behind each treble stitch) that looks like the picture, then it should work. I think the background stitches should be at least one stitch height shorter than the stitiches creating the X (or trellis pattern) on top. I just randomly chose a dc pattern with treble stitch on top. You might need to do some fudging to see if it works. But from the photo, it looks like the treble stitches make a sort of "spacing" in the background dc pattern, so you might want to avoid doing any double crochets in the stitch that has all of the treble crochets joining in it. So you do one double crochet per stitch, but the treble stitches share a stitch with the other treble crochet stitches.
Sorry for the rambling. I hope this helped.
Of note, I went to Annie's Attic website and looked under Community for their link called "stitching help" and saw what the Back cross-stich and Front cross-stitch looked like- and then I just improvised from there.
I wish I could video me doing it so you could see what I made. I don't think still pictures will help.