The five days mentioned pass before the jump is made.

Donovan felt as haggard as he looked. "Only five days?"
"Only five days. I'm sure of it."
Donovan looked about him wretchedly. The stars through the glass were familiar but infinitely indifferent. The walls were cold to the touch; the lights, which had recently flared up again, were unfeelingly bright; the needle on the gauge pointed stubbornly to zero; and Donovan could not get rid of the taste of beans.

_{"Escape!" by Isaac Asimov (emphasis mine)}

Then they experience the jump and its peculiarities. Afterwards,

(...) Powell pointed a trembling finger at the single gauge. The needle stood stiff and proud at the hairline whose figure read 300,000 parsecs.

_{"Escape!" by Isaac Asimov}

So the five days passing is no evidence for time dilation, since it takes place before the interstellar jump.

Those five days pass before the jump is made.

Donovan felt as haggard as he looked. "Only five days?"
"Only five days. I'm sure of it."
Donovan looked about him wretchedly. The stars through the glass were familiar but infinitely indifferent. The walls were cold to the touch; the lights, which had recently flared up again, were unfeelingly bright; the needle on the gauge pointed stubbornly to zero; and Donovan could not get rid of the taste of beans.

_{"Escape!" by Isaac Asimov (emphasis mine)}

Then they experience the jump and its peculiarities. Afterwards,

(...) Powell pointed a trembling finger at the single gauge. The needle stood stiff and proud at the hairline whose figure read 300,000 parsecs.

_{"Escape!" by Isaac Asimov}

So the five days passing is no evidence for time dilation, since it takes place before the interstellar jump.