What hard skills do I think new graduates could improve upon or demonstrate? Running larger-scale experiments (>100 grams of starting material) is always a nice demonstration that you understand the difficulties of scale-up. At the very least, telling people (photo in a presentation?) that you've used an overhead stirrer of some sort is helpful. Familiarity with basic analytical chemistry instrumentation is really important; that you can discuss the in-and-outs of fixing a HPLC that's beepingbeepingbeeping at you is a sign that you're someone that's adept at troubleshooting instruments. (This is a valuable skill anywhere, but especially at a small company.) A simple understanding of what a quality control laboratory does (and what it doesn't) and what makes a worthwhile certificate of analysis (and what does not) is more than I could have said for myself 10 years ago.
What soft skills do new graduates need to demonstrate? This is harder, a lot harder. My guesses: your advisor compliments you on a good notebook and good documentation of experiments. Being able to navigate the here-there-be-dragons aspect of reading through old texts, reviews, trade journals and patents and synthesizing it into a "OK, this is the subfield, and here's who the leaders are..." 5 minute summary is key. I like Keith Watson's comment on how to do this while still being in academia:
Industrial chemists can expect to work on dozens of technologies during their careers. Although a certain amount of mastery of a single discipline is needed to complete a dissertation, it is important that potential industrial chemists demonstrate that they are willing and able to learn new technologies. There are several ways to demonstrate this competency to potential employers, including learning and mastering the research of other professors within one's department. Another approach is to learn a new area of research every 6–12 months. This can be accomplished by investigating and reading the leading literature in the area...The problem is this: after 24 hours of sitting and thinking about hard and soft skills that younger chemists need, I've come up with 2 paragraphs of mush that don't really amount to much. Readers, I'm sure that I'm just not thinking straight, and there are skills that new graduates don't have, but should. I'd love to hear what you have to say.
[For a much lightyears-better version of this post, check out Derek Lowe's "Lessons for a New Medicinal Chemist". While it's (obviously) aimed at med chemists, I think it speaks to the conceptual mindset that is needed.]