Building a referral business for a pet care provider

Consider putting in a formal referral program in place

Encourage people to refer you and they get X% off the next time. 

Also I have also seen companies successfully use a discount off the quoted price to encourage a referral. 

You could say to them, 

“BTW there is a 30% for you if you can refer me to 3 people" (which means pass on their info, or send an email to that person and CC you with an intro). 

Nowadays referrals need incentives.

 Ask them to share (and make it easy for them)

Ask your clients to share your message on social media. Give them the blurb ready to go, “your story in 1 paragraph”. So all they need to do is copy and post to their facebook. Make it easy for them to do it. If they are happy customers and loved your service they should have no problem doing this. 

 A contest

Or run a competition. Ask your old clients, even your facebook page audience, and tell new clients that they could be in with a chance to win a free $150 groom for their pet. All they need to do is like and share your post and they are in the contest.

Best of luck with it.