This is a really interesting question.

The US Gov't's power to keep time falls under Article 1 Section 8 Clause 5 of the US Constitution:

To coin Money, regulate the Value thereof, and of foreign Coin, and fix the Standard of Weights and Measures;

Currently, the US Department of Commerce National Institute of Standards and Technology (NIST) is the regulatory authority for Weights and measures. However, traditionally, the US Gov't has taken a very hands-off approach to weights and measures and allowed industries to standardize themselves. (see, e.g., IEEE SA, ASHRAE, etc.).

However, every state also have its own weights and measures programs. This is somewhat because of the hands-off attitude.

Now, for time. The two main time standards are International Atomic Time (TAI) set by the International Bureau of Weights and Measures and the UTC (Coordinated Universal Time), another international standard computed based on observations of distant quasars. UTC is now standardized against TAI with leap seconds. The UTC (at the time Greenwich Mean Time or GMT) was adopted by the US in the Standard Time Act of 1918 (now in 15 USC s 261).

In the Standard Time Act, the time-zones of the US were established as were the standards used:

In this section, the term “Coordinated Universal Time” means the time scale maintained through the General Conference of Weights and Measures and interpreted or modified for the United States by the Secretary of Commerce in coordination with the Secretary of the Navy.

So, from that the US adopted the SI unit of a "second". So the US Sec. of Commerce can modify the standard (through a rule-making procedure), but time is, by statute, maintained through the General Conference of Weights and Measures.

According the the USDoT:

The time zones established by the Standard Time Act, as amended by the Uniform Time Act, are Atlantic, eastern, central, mountain, Pacific, Hawaii–Aleutian, Samoa, and Chamorro.

Accordingly, states may request to change a Time zone. But there is not apparent process for setting a particular time.

The request has to come from the highest political authority in the area subject to the request. Meaning a governor or legislature, or, from local government, from a position similar to the Board of County Commissioners.

The political authority has to provide:

1. Certification that the request is a result of official action.
2. Contact info of the person requesting.
3. Supporting Info - supporting contention that the requested change would serve the convenience of Commerce.

Requirements:

The principal standard for deciding whether to change a time zone is the convenience of commerce, which is defined very broadly to include consideration of all of the impacts upon a community that would result in a change in its standard of time. Examples of some of these considerations that should be addressed in the supporting information are the following:

• Where do businesses in the community get their supplies and to where do they ship their goods or products?

• Where does the community receive television and radio broadcasts from?
• Where are the newspapers published that serve the community?
• Where does the community get its bus and passenger rail services; if there is no scheduled bus or passenger rail service in the community, where must residents go to obtain these services? Where is the nearest airport; if it is a local service airport, to what major airport does it carry passengers?
• What percentage of residents of the community work outside of the community; where do these residents work?
• What are the major elements of the community’s economy; is the community’s economy improving or declining; what Federal, State, or local plans, if any, are there for economic development in the community?
• If residents leave the community for schooling, recreation, health care, or religious worship, what standard of time is observed in the places where they go for these purposes?