501(c) does not end at subsection (3), but instead at subsection (29). Some sections are used more than others. While each of these are fairly broad sets of organizations, 501(c)(3) reaches the furthest. However, even in not reaching as far, these sections are still important and expansion.
For example, Labor organizations can be tax exempt under 501(c)(5) Labor, Agricultural and Horticultural Organizations. Country Clubs or Hobby clubs, with limited membership, may be tax exempt under 501(c)(7). Credit Unions organized correctly can be tax exempt under 501(c)(14) and Cemeteries under 501(c)(13).
While these organizations, and more, can receive tax exempt status, there are also some drawbacks. For one, there are additional annual reporting requirements such as the 990-series of forms. Additionally, the tax exempt organization must be operated only for the exempt purposes. This can get tricky as a Church selling flowers could be a business operation or it could be for the exempt purpose of supporting the Church. Additionally, the tax exemption is for public benefit and if private benefit is found there may be consequences.
There are more sections, as indicated there are 29 subsections, some with multiple exempt categories. When you are looking to start an organization it would be prudent to check with the IRS, a CPA, or a local attorney to see if your organization could be tax exempt. A brief statement such as this cannot give you the full understanding of the tax field. These categories can be complex. Still, if your organization qualifies, it can open new avenues of operation, as well as limit some activities.