Should I offer a Quick Visit option?

My visits are 30 minutes long and I schedule my visits 30 minutes apart. Meaning if I have a client at 7, I schedule the next one for 8, even though it doesn't take me 30 minutes to travel to them. If I'm crazy busy, then I come at 6:30, but I can really only have like 3 clients at a time, unless I get cats where it doesn't matter when I come.
 
I did some cats last week that were hidey and the owner said I didn't have to stay for 30 minutes, but he paid me my full fee and I came twice a day at non-set times and felt pretty guilty about it.
 
I belong to a lead-generating service where clients put in their info, and I get a text and email with their contact info. It goes to at least one other company and you basically compete for the business. Usually, it's whoever gets to them first. Last week I got a cat client who is gone for two weeks and it doesn't matter what time I come.
 
She kinda hemmed and hawed when I gave her my rate, and I really wanted the job, so I gave a dollar a visit discount, which she accepted and I got the job. At the meet and greet, she said that my competitor was offering the service for 2 dollars cheaper than my discount, but that she would still use me. The cat is 19 and sleeps 24/7 and only likes the owner, so again, she said I didn't have to stay for 30 minutes.
 
I'm wondering if I should offer quick visit option for clients who are looking for a bargain. I still won't charge rock bottom pricing, but maybe up to 2 dollars cheaper a visit. I could take more clients and get more money in the long run and many of my clients will still opt for the 30 minute visit, since they want their pets to have some human companionship while they are gone. I could take on some more mid day clients, since I get requests to "just let the dog out back to pee".
 
What do y'all think?
 

Edited because this is a family site and Karina made me think impure things about the phrase "In and Out".