Do you necessarily need to wait one year before attending a PhD in the US?
In all phases of a US student's career, it is understood that many people will be applying based on their anticipated graduation date, rather than having already graduated.  Thus:


High school students apply for colleges in the fall, even though they won't graduate until the end of spring.
Undergraduates apply for graduate schools in the fall, even though they won't graduate until the end of spring.
Ph.D. candidates apply for postdoctoral positions the fall or early spring, even though they won't graduate until the end of spring.
Postdocs apply for faculty positions in the fall, even though they typically have contract commitments through the spring or summer.


This does leave a problem of what can happen if one fails to actually graduate as anticipated.  It is, however, generally the case that technically offers are made contingent on the current phase of one's career completing as anticipated.  Usually this is a fairly safe bet, since it is unlikely that a strong candidate will suddenly plunge in performance so badly that they cannot graduate.  In some cases, however, it will happen and in that case either:


The offering institution may with withdraw their offer, or
The offering institution may allow the candidate to defer for a year, or
The offering institution may allow the candidate to start anyway.


The second is much more likely to happen in cases where the unexpected failure to progress is due to external circumstances, such as serious illness or having a child, which would not cause an institution to reassess one's potential, and the third where everything is done except some final formalities (e.g., a graduate student has defended their thesis but starts a postdoc before depositing).
The chances also depend on the stage of one's career: there is often a great deal of flexibility for faculty start dates, for example, while postdocs are often tied to specific funding with a specific non-deferrable timeframe.