Moving a small website with 10 pages or so to another host shouldn't have any effect on the site's ranking in the SERPs unless there is a significant difference in the plans or site structure. Regardless proper 301s, canonicals, etc. are a requirement.
Site Speed
Site speed is an important factor in ranking well in the SERPs. A host to deliver a site faster if it is located in the same country as the requester, but the number of websites on a single server makes more of a difference. The cheapest website hosts place as many sites as possible on a server. For smaller websites that do not make a number of database requests or that do not receive significant traffic, this is not that important.
Host Location
Moving to a host located in the same country as the website's target audience will help SEO, assuming the previous host was located in a country that the website did not target and no targeted country set in Google or Bing's Webmaster Tools. This does not apply to websites with a worldwide audience and multiple language options.
Uptime
Almost all website hosts, even the most affordable ones, guarantee 99.9 percent uptime. Being down not only annoys visitors who may not return, it prevents the search engine's bots from crawling a website.
Bad Neighborhood
A website host that hosts a large number of gambling and pornography sites may derail the SEO efforts of sites not in these sectors. The neighborhood does make a difference.
Switching hosts will not hurt a website's SEO efforts, unless one of these four factors occurs, and the change is drastic. Reading reviews of website hosts before making the decision to switch will help avoid preventable problems. Cost and customer service also matter, no one wants to overpay or not be able to reach a customer service representative when their website is down.