Since the data offered at IPUMS-USA is public-use microdata, certain steps are taken to preserve the anonymity of respondents. Among these steps is the suppression of geographic information at lower levels. In the case of the 1960 and 1970 samples, state is the lowest level of geography that is made available. Because of this, migration variables for 1960 and 1970 are more limited than in other years. In 1970, the migration variables that indicate the past residence of respondents are MIGRATE5and MIGPLAC5, which report whether a respondent has changed residences the last five years and if so, which U.S. state, outlying territory, or foreign country they moved from. Unfortunately, MIGPLAC5 is not available for 1960, which means it is not possible to identify from which state or country a respondent moved. You can however, identify whether their previous residence was a metropolitan area through the variable MIGTYPE5.
For information at a more detailed geographic level, the only possibility is to view it in aggregated form such as the published census tables you mentioned in your original question. You may also find the NHGISproject to be a helpful resource, as it offers many published census tables going back to 1790 at several geographic levels.