It is a bad idea to build Qt yourself in general, and in this case, it is not even warranted to have a static build. You have several ways to solve this issue:
From the documentation:
All Qt programs automatically support the following command line options:
\list
\li \c{-platform} \e {platformName[:options]}, specifies the
\l{Qt Platform Abstraction} (QPA) plugin.
Overridden by the \c QT_QPA_PLATFORM environment variable.
\li \c{-platformpluginpath} \e path, specifies the path to platform
plugins.
Overridden by the \c QT_QPA_PLATFORM_PLUGIN_PATH environment
variable.
and
void QCoreApplication::addLibraryPath(const QString & path) [static]
Prepends path to the beginning of the library path list, ensuring that it is searched for libraries first. If path is empty or already in the path list, the path list is not changed.
The default path list consists of a single entry, the installation directory for plugins. The default installation directory for plugins is INSTALL/plugins, where INSTALL is the directory where Qt was installed.
See also removeLibraryPath(), libraryPaths(), and setLibraryPaths().
Depending on your exact desire, this could be solved one way or another like that.