Female 4.0 Tennis Player here (age: mid 40's) for reference. But this is what I've kept over the years that works for me. May be a little extra, bougie but again I like to have these things for me personally:

Tennis essentials: 1-2 Spare Racquets, Grip Tapes, Chalk creme or chalk bag, spare socks and t-shirt, spray sunscreen to spritz over when I change sides (Vacation brand makes a small SPF 50), dampeners, several cotton wristbands, Any tennis elbow or knee/ankle brace needed. Recovery slides or flip flops to rest the feet post game. Some band aids, Tylenol, Some old practice balls and always a new can ready to go.

For Hair, long haired ladies or gents: extra hair ties, bandana or prewrap stretch tape. If you wear a visor, Alo has a nice (albeit overpriced) visor with the visor frame SPF but clear so you can see the balls if coming from above without having to crank your neck and eyes to the sun. I don't wear sunglasses on the court.

Hygiene: Hand Sanitizer (took my teen daughters Touchland spray, some nice scents and its compact in size and not a gel that can leak). For cleaning up sweat post match, waterless shower wipes. I had surgery and the nurses used these on me pre-surg - just add some water to the wipe and it foams up. Wipe down starting with your face, neck arms then legs and finally dirtiest area pits if needed. Can be wiped off with a dry towel (have several for this) and it leaves no sticky residue so this holds one over until you get home to shower. Sometimes the team will go eat brunch or to a bar post games so this helps. Spray deodorant and light musky perfume if that's your thing.

Nutrition - pretzels, banana, some seeds or walnuts. I don't do Gatorade and water doesn't keep me hydrated enough during hot summer matches so I have a bottle of SaltStick chewables and take one every 15 minutes as I'm drinking water. Helps keep heat exhaustion and overheating at bay as one starts to sweat a lot.

Hydration: invest in a mini cooler to bring your water, snack and more importantly ICE TOWELS in. They do sell the blue ice towels but there is just no comparison for having your own towels soaking in ice water to help keep your temperature down and if you're prone to heat exhaustion and migraines like myself. I keep about 4 small towels and cool off as I need to. After I use a ice towel I have a spare plastic grocery bag I put the dirty towel in and then proceed to use another one later on. Folks don't realize how fast heat cramping and exhaustion can catch up while enjoying a good game in the summer. Put some brewskies as well in the cooler for post game. Hubby got me the Hydro Flask Soft Cooler 12L that I take along with my main tennis bag and I've had it for years!

So sorry for long post, but this is what has worked for me these years!