If a unit is fully ready and I already have a tenant lined up, I will allow them to start moving in a few days before the official lease date without any extra cost (assuming I have the security deposit and first month's rent, in full).
Now that I'm thinking about it, I do a lot of freebie gifting!
For example, I have a tenant who is partially living and partially using my rental house as an AirBnB (I knew that ahead of time). The previous tenants had moved out, but only took their clothes. I took the few things I wanted and actually threw out or put on the curb most of the other stuff before I did showings. But I'd kept all the plateware and pots/pans in the cabinets, jic the next tenant might want them. He was SO thrilled I'd done that! All of it was stuff he would have had to buy anyway.
I also had a win-win with a tenant last year. She was moving in, but didn't have any furniture. Coincidentally, an evicted tenant had moved out a few days prior and left their living room set and large tv armoire. We were just going to throw them out/leave them on the curb. But I told the tenant she could have them, if she wanted them, and be responsible for moving them out. We saved them for her for a few days and made arrangements to meet her with brother and dad at that other unit. She wanted the furniture items and was ecstatic to get them for free. We were happy to have someone else take them away!
I had a set of tenants who were in the process of buying a house. They ran into a lot of bumps and their closing kept getting pushed out, even though they had already given me a 30-Day notice they were leaving. Fortunately for them, I was planning to redo the kitchen when they moved out, so I wasn't looking for a new tenant yet anyway. I allowed them to shorten the notice to only 14 days because their closing had become such a moving target.
As it turned out, for their last notice when all the ducks were in a row and they were definitely moving out on X day, they were going to be a few days shy of the two weeks. They asked if I would forgive the prorated rent for those days. I didn't have to and they understood that. However, they'd been fantastic tenants and I agreed to it. It was a loss of about $160, but I've never regretted that decision. In my own heart, I felt it would have been greedy to have insisted on it just because I could. To me, it wasn't a real loss because it only meant I started on my kitchen reno four days earlier.