Are you talking about actual design mockups (PSD files etc) or just directional sketches that will help illustrate ideas you're having about how to improve the pages?

If leaning more towards the former, I'd put yourself in a position to not need anyone else necessarily and just focus on creating wireframe mockups of the pages (no typography or finalization of icons/the skin of the page) and start with the blocking and tackling of each page. wireframing it also let's you more easily bifurcate between the web and mobile UX (if that is a part of what you've identified as an area that can be improved upon) - plus in my experience wireframes give you more of a blank canvas to start with while not narrowing your possible ideation phase with already in-place icons or imagery - basically it lets you work from the feet up each page. Taking this step (with wireframes first) allows you to revise and iterate on your ideas and THEN go find someone to produce mockups/design files in a more streamlined fashion (you dont want to be creating and constantly editing full design iterations as that would be a heavier, cumbersome and most costly process.) I've also found the narration process around wireframes much smoother and easier for inviting other contributors and capturing the commentary of the "why" to go with your "what".

Good luck, feel free to reach out if you need any other suggestions or walk to talk through options!

Send me a URL for what you want to improve upon, I'll check it out and give you some other resources if I can't help.

best,

Lee

Direct Answer:
You can't really get cheaper than Upwork or Fiverr or Craigs List. And you'll get what you pay for.
Since you have a starting point (your competitor's site), you could do it yourself on paper: print it and draw your edits, then give it to your front-end dev. (with their code pulled from their page, if you need to).

Longer Answer:
I deal with a lot of startups that hope to build a digital product that's a "better" solution.
I'd caution you: only IF and AFTER you have figured out on your own what your potential customers/users need, and have concluded that the compeitor's site meets (some of) those needs (even if partially) should you use a competitor's actual visual UI/UX design as your starting point Otherwise, you'll just expend effort to make their poorly fitting design less poor. To get and keep customers, you need a GREAT UX, not just a better UX. No customer is happy with a poorly fitting UX that fits their needs less poorly than another business.
Analyzing your competitor's product/service to make your own design fit your customers' needs well, is smart. Copying their design, might not be.

If this helps and you want more guidance and specific design advice, get in touch here about having a short intro call.