"during the time" is a correct and usable phrase in written English. It can be used to refer to an unspecified length of time in the past, present, or future. For example, "During the time that we have known each other, we have been through a lot together."
One Rape, Please (to Go) .
"It was during the time of the hunger strikes.
The opera boxes during the time of the great exhibition!
He was in grade eight during the time of Partition.
Of course, that changed during the time we lived there.
That aqueduct was begun during the time of Augustus.
Then, during the time trial race, he crashed.
I remember the little towns of eastern Kenya Manderara, Garissa — during the time of the Somalian war.
Rebecca Spang displaying the different and distinct local French currencies during the time of the French Revolution.
Except during Summer Quarter, you must be enrolled during the time you live on campus.
"Michael [Forsythe] both reported and wrote that story during the time he worked for the Times.
Ludwig does not simply clarify my doubts with English writing, it enlightens my writing with new possibilities
Simone Ivan Conte
Software Engineer at Adobe, UK