US nuclear weapon production in the postwar was initially very slow, for a variety of bureaucratic and cultural reasons. So the US stockpile grew initially quite slowly, and was limited until about 1950 to essentially the same kinds of bombs that were dropped on Nagasaki (and a handful of the Hiroshima type). During this time, the fuel for the bombs was produced by the civilian Atomic Energy Commission (AEC), per the Atomic Energy Act of 1946, and could not be transferred to the custody of the Department of Defense (DOD) without the explicit authorization of President Truman.

Truman was very uncomfortable with the idea of the military having physical control over atomic bombs, because one of his take-away interpretations of World War II was that the military treated nuclear weapons as just "another kind of weapon" and not as "special" or "political" as he did. So despite the DOD urging him to transfer weapons to them for stockpiling either in the US or abroad, Truman kept the cores in civilian hands through most of his presidency. So, to get to your question, they were in this period being stockpiled, but even then they were split between the cores (held by the AEC) and the rest of the bombs (held by the DOD).

The first exception to this came in 1951, when, during the Korean War, Truman authorized that nine nuclear weapons could be transferred to an aircraft carrier near Guam, under strict orders that they could not be used without Presidential authorization. This was not only the first deployment of US weapons abroad, it was the first deployment of US nuclear weapons to the military since WWII.

When Eisenhower became the President in 1953, his attitude was pretty different, and the world was rapidly changing in some key ways. Eisenhower was more worried about the possibility of an "out of the blue" attack by the Soviet Union than he was about the military "jumping the gun" on nuclear war, and the world was rapidly entering into an age of thermonuclear weapons and missiles. A wide variety of new nuclear weapon designs, and a nuclear arsenal that was being measured in the thousands of weapons, and a world in which the USSR had nuclear weapons, meant that the civilian-military split enforced by Truman was no longer being seen as quite as practical, and Eisenhower began the transfer of many thousands of these weapons to the DOD. He also began deploying these weapons abroad, so that the US would be able to start nuclear war in a matter of minutes, rather than a matter of days or weeks. So this included a flurry of deployments in the following places outside of the continental US: Hawaii (1954), Okinawa (1954), Morocco (1954), the United Kingdom (1954), and West Germany (1955). These were deployments mainly of bombers and gravity bombs, but also short- and intermediate-range missiles, as well as tactical nuclear weapons. In some cases these were pretty publicly known and not surprising (like the United Kingdom and West Germany), but some of these deployments were secret until relatively recently (like Morocco).

There were few more overseas deployments in 1956 (Alaska, Puerto Rico, and Iwo Jima), but Sputnik and the "shock" of confidence it created in the United States and its allies abroad really ramped up the deployments in 1957–1958: France, Greenland, Italy, Philippines, Spain, South Korea, Taiwan, and Tunisia.

(If you would like to see these deployments from 1945-1958 as a map, here you go. This map is just the deployment of bombers and bombs to overseas US bases, not tactical or missile weapons, but the overlap between those deployments is pretty high. There were also lots of bombers without nuclear weapons deployed to bases, there were also bases that specialized in other kinds of attack support — like mid-air refueling — that are not shown here. So the actual "encirclement" of the USSR/China was quite larger than even this one. Over time, some places gained nuclear weapons bases that didn't have them before — like Turkey and the infamous Jupiter missiles put there in 1961, and at some point the Netherlands became a host for NATO nuclear weapons sharing — and some places removed them.)

So you can see that by 1958, the US had deployed these weapons in a "ring" around the Soviet Union and China, ready to deploy at a moment's notice. The bomber-based war plans at the time essentially foresaw being able to destroy the USSR, the Warsaw Pact, and China (the war plans of this period did not allow the US President not to destroy all of these together) in a matter of hours. There were also (not included in the above) deployments on sea-based platforms like aircraft carriers (and a few, sad early submarine-based missile efforts — the Regulus submarines, which were basically the Navy attempting to have some kind of submarine deployment while the Polaris program was under way). These would be augmented by some early (and not super reliable) missile deployments (like the Snark and Thor intermediate-range missiles, deployed in the UK in 1959). Some of these were formal NATO nuclear weapons sharing agreements, but many of these were not — and some were so secret that their presence was kept hidden even from the governments of the states that were hosting the bases where these were kept (this was the case for Denmark and the deployments in Greenland, and Japan with the deployments in Okinawa and Iwo Jima; in both cases, some of this was a case of them not wanting to know, because plausible deniability was felt to be a better situation than the possibility of a broader public debate about the policy).

These bases abroad were both an attempt to reassure allies (especially in Europe) that the US was willing to use nuclear weapons in the case of a Soviet attack against them, but also a key part of the technology that the US primarily focused on for delivering these weapons at the time: bombers. While the US did develop bombers that had intercontinental range (thanks to in-air refueling), its war plans anticipated a rapid ability to penetrate Soviet airspace simultaneously from basically all sides, and this required having the bombers and nukes very close to this airspace.

It is of interest that the US initially assumed that the USSR would follow a similar technological path, despite not having (until the Cuban revolution) allies near US borders. Instead, the USSR focused on long-range missiles — part of the Sputnik shock (the rocket that put Sputnik into orbit was also one used in first Soviet ICBMs).

Anyway, there is more that can be said about this. As you can see there are technical, political, and strategic aspects to this issue. I have written a lot about early US delivery system development, as well as some more discussion of the so-called "custody dispute" over the atomic bomb (the AEC and the DOD split).