I don't think there is a generic answer to this question, I don't even think there is an answer at all. It depends a lot on the type of work the company you work for does.
I have found myself in exactly the same situation on several occasions, particularly working as mentioned in the question: sole designer with several developers.
I can only give an answer from my personal experience, about what I did at the time. Not all options are very creative, I explain it better in the following points.
Avoid self-flagellating with inspiring creativity :-)
A basic error in behavior that all designers tend to have is that our work must be 100% creativity all the time. This, taken to the extreme, is quite unproductive, it leads us to believe that no task outside of creativity is valid, causing situations of anxiety, nullity and doubts about our role in the project. The question statement says - how to stay inspired, motivated and active –. I think it's too demanding an approach, both on a company level and on a personal level. I'd trade all three for one simple action: how to stay productive. This may not be inspiring or motivating, but it's activity and at least productive for the company and the designer.
Clean up / Arrangement / Back up
This is the typical tedious process that no designer wants to do at any time, so leisure time is key to doing it. In general, the designers' file folders are usually full of the "final file", the "better than final file", the "definitive final file", the "definitive2", the "definitive plus top file", etc, etc, etc ...
My approach was based on thinking that my work would pass into the hands of another person, especially being the only designer. This hypothetical person should sit in front of my computer and clearly find each file and immediately know how to interpret its purpose.
At the end of a project we usually abandon it definitively due to saturation. Seeing ourselves 100% involved in something certain for a while makes that later on we don't want to know anything about it, or see or analyze it, much less review the graphic part. However, it's a very good perceptual exercise. I usually call it making "skins" of old designs: imagine, take notes or directly make prototypes with new graphic options of past designs, the older the better. I insist, it's a very good perception training and it helps to keep the creative head always fresh.
Creativity not only manifests itself in the designated work, but also possible ones. It often happens to designers that while we are working on a project our mind travels through possibilities unrelated to it. A typical example, developing the interface of an application, we usually think about how the animated description of this project would be. Well, the time to zero activity in the company is the right time to develop that parallel project that we had in mind. This is not only useful for the designer, but it can also be useful for the company or the project. An animation of the type of project, its use, its possibilities, etc. They can even be used in the future either as an improvement, implementation in another project, presentation to clients, etc. And it doesn't have to be a Pixar movie, just open Keynote, Powerpoint or a simple html and visually unfold the step by step. It's very clarifying for all parties: the designer, the developers and the company itself.
I don't develop it since it is included in the question, but the study is never too much, especially now that new tools appear every day.
Research on the internet about graphic design contests, there are several resources and calls on different topics. If you find something interesting, the company may well be interested in participating or perhaps allow the designer / worker to participate on their own, mentioning who the work was intended for.
Well, sure there a more options, but my personal opinion is that believing and understanding the first point will help to find more activities than the ones I describe in this answer.
By the way, I think this question is off-topic in UXSE ;-)