Do any other Hindu scriptures mentions similar advice?
Yes there are scriptures that ask them to do so.
"Outside the rains let him spend one night in a village and five
nights in a town. During the rainy season, however, let him live in
one place for four months [ArU 8, n. n].
"A mendicant should not spend two nights in the same village. Doing so will give rise to passion and the like, as a result of which he
will go to hell. "In deserted areas on the outskirts of villages let
him roam the earth like a worm, homeless and controlled. During the
rains let him stay in one place.
From the Narada Parivrajaka Upanishad's Chapter 4
Only during the 4 months of the rainy season a Sannyasi is allowed to stay in a fixed abode. But during the other months he should be constantly on the move.
Some more examples are given below:
A wandering ascetic should wander forth according to the rule,
abandoning his relatives and free of possessions. 17 Going into the
wilderness, 18 he has his head shaven except for the topknot; 19 wears
a loin cloth; 20 resides in one place during the rainy season 21 and
wears ochre clothes.
Badarayana Dharma Sutras 2.11.26
Now then the duties of the highest class of itinerant monks – the
Paramahamsa Parivrajakas (are as follows): They must sit and lie down
on the ground. Those having already taken the vow of chastity etc.,
should use an earthen bowl or one made of gourd, or a wooden bowl;
they should give up lust anger, avarice, infatuation, ostentation,
haughtiness, jealousy, attachment to objects, egotism, falsehood and
the like. The Sannyasin should stay at one place during the four
months of the rainy season and during the remaining eight months
wander alone, or with a single companion, aye, a single companion.
Aruni Upanishad Chapter 1
As regards your other question viz- "Why is he required to do so?" - the following passage, which are words of Lord Brahma, gives some reasons:
The Grandfather, after eulogizing Narada who asked about the
observance of ascetics, replied thus : The ascetic that has
attained indifference (to objects), should stay in one and the same
place in the rainy season (for four months), and then for (the
remaining) eight months should wander alone. Then also the ascetic
should not stay in one and the same place for more than a day. Like a
deer that does not stay in one place on account of fear, he should not
stay in one place. He should not create an attraction (in his mind)
that may serve as an obstacle to his going about. He should not cross
a stream (by swimming) with his hand, nor ascend a tree, nor witness
the festival of a God,
Narada Parivrajaka Upanishad's Chapter 7
So, what i have understood, is that by staying in one place for more days, attraction, attachment, passion etc may rise in him and also a sense of rest, and all these are considered as obstacles to his spirituality.
So, that's why he should always be on the move.
NOTE: 1. The Narada Parivrajaka Upanishad is linked to the Atharva Veda and Aruni to the Sama Veda.
2. There are other Upanishads as well, that have similar instructions.