There are quite a few things going on in this question, I would simply break it down into a two-pronged approach:
1. The demands of the audience and expectations placed on either Shakespeare or Atwood. In a simple sense, composers attempt to use "mass" literature, that is the forms that are consumed most commonly by their audiences. This ensures the focus of the text is not arguing why the form is the best mechanism to share meaning, but rather, actually focus on the core ideas within the text. (Easier to accept the meaning rather than audiences trying to "understand" and figure out a new form - that may be unfamiliar).
That is, if Atwood reappropriated Tempest into another play, it would't perceptively connect to audiences as well as the novel does. This is because we as a modern audience want more intellectual autonomy in understanding the characters, but also in a capitalist sense - novels sell the most at present and is what is more commonly published. (e.g. Hogarth Press)
Hence, the reason I would structure my argument on the basis of the changing demands of the audience is that it doesn't influence the form but rather forces composers to use a particular structure and form. This is quite an interesting argument about authorial intent and do composers actually have a sway on the base form. You can explore and delve into it a bit more.
For my second aspect to this response:
2. How has the successor changed my impression of the hypotext (edit from hypertext).
This is quite a personal opinion and this question really invites you to consider what you think Hag-Seed has showed you and what meaning you have achieved from reading it.
For me, I've learnt that the abuse of power is something that people so readily accept and has become an engrained aspect to how our society operates.
The "acceptability" of revenge and individual power compasses has been reasoned by Shakespeare through the degradation of the Great Chain of Being while Hag-Seed operates on the destruction of our strongly held meritocratic values.
In order to explore how Hag-Seed has "thrived" is that it shows the nuances that we as modern readers don't see or realise on a daily basis. Though the abuse of power remains constant, how it is embedded within our daily lives has significantly changed - this is where Atwood thrives.
An example of this is how Felix promises an "early parole" to 8Handz that demonstrates our "acceptance" of this abuse of power because it benefits both individuals. At first reading, I was quite supportive. However, after understanding more deeply Felix's intentions - he can control 8Handz, while 8Handz will be returned back to the "utopia" of the outside world (something he longs for). Though this benefits both parties - it unfavourably favours one over the other.
This has allowed me to re-analyse Ariel's imprisonment by Prospero as through my modern-lens it seems a gross violation of human rights, but after reading Atwood's text, I have realised that Prospero has actually "freed" Ariel from Sycorax - he isn't physically constrained and he is actually considered a "saviour". This is because he is re-instating the Great Chain of Being.
Therefore, in the same way as we accept Felix's control over the prisoners (that is quite nuanced and very deliberate choice by Atwood), I assume, the contemporary audience of Shakespeare similarly accepted and believed in Prospero's control over Caliban and Ariel.
This is one example of how Atwood has thrived - it has made me realise that the abuse of power is more and more common as the capitalist system promotes a decollectivitised society - one that focuses on individual gains (hence manipulative abuses of power).
This is just one way of approaching the question, but again remember to provide a judgement on "extent". If you have any questions feel free to ask.