Addressing Employment Gaps in Your Resume
This is a common phenomenon faced by candidates who quit jobs either to join other firms or to pursue other ventures. When this fails, they land in a job search and face such situations. My suggestions are:
Ensure the facts are clearly mentioned in your CV, the timeline is properly tied up, and they are adequately verifiable. If possible, one should produce enough convincing documents in support of the facts. Without documentary support like letterheads, visiting cards, or business tie-ups mentioning your name, a relieving letter from the existing business entity will add to your claims.
Likewise, you should mobilize the support of an existing employee of the prospective firm, preferably in a senior position, to recommend you. On the contrary, if you advance this stand just to fill the gap in your resume, it won't work that way. Even if you secure the offer, you may get caught when the background verification and credential (BVC) report is out, in case it reveals otherwise.
So, give it a try through your past colleagues or friends in employment, even if it might not meet your expectations. If you can prove that you are the candidate they are looking for, you won't fail in your attempts. Not all employers view this situation in the same manner.