To increase chat engagement, have the chat bubble say the following when visitors first land on your site:
"Thank's for your interest in [company's] [product], can I give you a tour of [whatever it is you want to engage with them on]."
The general idea is to 1) thank them for giving you something (e.g. their interest in you and your product), and then to 2) offer to give them something (e.g. a tour of xyz).
If you'd like an explanation for why this structure works, or suggestions for structures, designed around your specific product, let me know,
I think you have two options or can use a combo of both. Without knowing more about your business or site, it's difficult to be very specific, but here goes...
1) Make sure the chat feature is clearly visible from every page, but not obstructing normal navigation.
2) Program live chat to pop up at key points in the buyer's journey. For example: If someone is lingering too long on a page, you could ask if they need help. If they are abandoning their cart, you can try to redirect them. You could even use live chat to suggest a complimentary item.
However, if you find that you need live chat in order to complete the majority of your sales, you may need to reassess your website to ensure it's doing the heavy lifting you need.
1. Make sure that the website - the content and design - is engaging. People should want to ask for more information, not ALL information.
2. Be online regularly. Constantly being offline is frustrating. Regular visitors will notice this.
3. Assess their voice and adjust yours as appropriate. Speak simply and provide good examples and links.
4. Never let Live Chat be a replacement for detailed product and service descriptions. It's an addendum.
I would also recommend adding Facebook Messenger as a live chat option. So many people use it and it builds trust with your brand in a medium they're familiar with. Try this integration [disclaimer - I've never tried it]. https://www.livechatinc.com/facebook-messenger/
Secondly, use the chat interactions as source material for your content marketing. Every question should already be an FAQ answer.
Also build a mailing list using the Facebook pixel from your Messenger interactions - you can do push messaging using FB direct into Messenger which is an awesome replacement for email marketing.
Happy to discuss further if you need.