spacing in the table's minipage

(this is a slightly expanded version of my earlier comment)

Don't use the fairly low-level \linespread command to change from single-spacing to, say, one-and-one-half-spacing. As you've discovered, it affects not only the main body of the text but also the material inside minipages, tabular and array environments, footnotes, and so on.

Instead, load the setspace package (ideally with the option nodisplayskipstretch) and then execute the package's \setstretch command, say, \setstretch{1.5}.

\usepackage[nodisplayskipstretch]{setspace}
\setstretch{1.5}

The setspace package also offers macros with fairly descriptive names --\singlespacing (equivalent to \setstretch {1.0}), \onehalfspacing, and \doublespacing -- as well as environments with these names.