Short answer: it depends on your understanding of biblical inerrancy. If you hold unqualified biblical inerrancy, then to be consistent you should hold that Abram left Haran after his father had died. If you hold qualified biblical inerrancy, then you can hold that Abram left Haran either while his father was still alive or after his father had died. Long answer follows.
The biblical text supports the notion that Abram left Haran after Terah had died in two ways:
1. The fact that Abram's call and departure come after Terah's death in the text
Terah took Abram his son, and Lot the son of Haran, his grandson, and
Sarai his daughter-in-law, his son Abram’s wife; and they went out
together from Ur of the Chaldeans in order to enter the land of
Canaan; and they went as far as Haran, and settled there. The days of
Terah were two hundred and five years; and Terah died in Haran. Now
the LORD said to Abram, “Go forth from your country, and from
your relatives, and from your father’s house, to the land which
I will show you;[...]” So Abram went forth as the LORD had spoken to
him; and Lot went with him. Now Abram was seventy-five years old when
he departed from Haran. Abram took Sarai his wife and Lot his nephew,
and all their possessions which they had accumulated, and the persons
which they had acquired in Haran, and they set out for the land of
Canaan; thus they came to the land of Canaan. (Gen 11:31-12:1,4-5)
Evaluation of strength: The relative placement of Terah's death and Abram's call in the text does not necessarily imply their relative placement in time, since the human author may have just wanted to round up his coverage of Terah's life before moving on to the events in Abram's.
Besides, the divine call to Abram to "go forth [...] from your relatives" fits much better with Terah being still alive at that time, because, if Terah was already dead, and given that Abram took Lot and Sarai with him, who were those "relatives" of Abram whom he was called to go forth from??? (Note that Terah had taken only Abram, Lot and Sarai with him when he moved from Ur to Haran.)
Along the same line, the divine call to Abram to "go forth [...] from your father's house" fits better with Abram's father being still alive at that time.
To note, an alternative view  posits that the call to Abram in Gen 12:1-3 occurred in fact while Terah and Abram were still living in Ur, but was placed in the text after Terah's death as a flashback, to connect it with Abram's much later fulfillment of that call. The problem with that view is that, according to it, Abram was called by God to go forth from his relatives and from his father's house but fulfilled the call only when there were no longer any relatives or father to go forth from!!! (Despite the fact that the text says that "So Abram went forth as the LORD had spoken to him;")
2. Deacon Stephen's statement on the issue in his testimony to the Sanhedrin:
And he said, “Hear me, brethren and fathers! The God of glory appeared
to our father Abraham when he was in Mesopotamia, before he lived in
Haran, and said to him, ‘LEAVE YOUR COUNTRY AND YOUR RELATIVES, AND
COME INTO THE LAND THAT I WILL SHOW YOU.’ “Then he left the land of
the Chaldeans and settled in Haran. From there, after his father died,
God had him move to this country in which you are now living. (Acts
Evaluation of strength: First of all, there is the issue of whether the doctrine of biblical inerrancy requires one to hold that Stephen's view on this particular issue is correct. This is clearly not the case, as what biblical inerrancy requires one to hold is only that Stephen's testimony before the Sanhedrin was faithfully recorded by Luke.
The freedom of Stephen's testimony from (relevant) errors does not derive from biblical inerrancy, but from Jesus' promise of the Holy Spirit's assistance to Jesus' disciples when brought to give witness of his faith before persecutors:
“When they bring you before the synagogues and the rulers and the
authorities, do not worry about how or what you are to speak in your
defense, or what you are to say; for the Holy Spirit will teach you in
that very hour what you ought to say.” (Lk 12:11-12)
It is this particular promise, and not the doctrine of biblical inerrancy per se, which brings Stephen's testimony under the umbrella of inerrancy: if the Holy Spirit was teaching him what to say, then his testimony must be free from any (relevant) error. But the issue facing the exegete (and whose treatment I anticipated by the previous 2 bracketed instances of "relevant") is exactly the same: does the inspiration of the Holy Spirit, either to the writer of a biblical text or to a Christian giving testimony before persecutors, guarantee that the text or testimony is free from errors at any and all levels of subject matter, or only at those levels of subject matter which are relevant to our salvation?
If one adopts the former position, which I call "unqualified biblical inerrancy", then, to be consistent, he must hold Stephen's testimony in the same way and therefore hold that Abram left Haran after Terah had died.
The latter position, which I call "qualified biblical inerrancy", consists of holding that:
the purpose of divine Revelation is to teach the truth which is relevant to our salvation, and not to teach natural science, profane history, or other forms of merely worldly knowledge for their own sakes, and
the errors or lack thereof (i.e. inerrancy) of a text (or of Stephen's testimony, in this case) are defined in relation to its purpose.
Thus, qualified biblical inerrancy allows for the presence of factual errors in the Bible in profane matters of no relevance for what Scripture properly intends to affirm.
Focusing now on Stephen's testimony:
What was Stephen intending to affirm when talking about God's call to Abram? Clearly that God's presence and action was not confined to the boundaries of the Land of Israel, by noting that He had appeared to Abram in Mesopotamia - meaning by that Ur, which was farther away from the Land of Israel than Haran, which BTW was also part of Mesopotamia - which was in line with Gen 15:7 and Neh 9:7.
How relevant is to that point the issue of whether Abram left Haran before or after Terah's death? Even more clearly, not relevant at all.
More generally, how relevant is this issue to faith in Christ? Most clearly not relevant at all, particularly in view of Lk 14:26.
So why did Stephen state that Abram left Haran after Terah's death? Probably because that was the prevalent understanding of the passage in the Jewish community of Jerusalem at that time, so he just adopted it to move on with his argument.