This is a good question—and I disagree with SovereignSun's view that "another three questions" is flatly incorrect.
From a fiercely prescriptivist viewpoint, one might argue that writers and speakers should never use the expression "another three questions" because "three more questions" expresses the underlying idea just as clearly and succinctly—and doesn't do violence to the essential sense of another as meaning "one other."
Idiomatically, however, people use constructions of this type all the time. Indeed, a Google Books search turns up multiple instances of the precise phrase "another three questions" in professionally published texts—including one on English grammar. From Lise-Lotte Hjulmand, A Concise Contrastive Grammar of English for Danish Students, fourth revised and expanded edition (2012):
184.108.40.206. Both, either, neither, another, other
These can all be used both as pronouns and as determiners. Both, either, neither and other relate too two. This is sometimes also the case with another, but this expression can also just mean 'one more, additional, extra'.
Another usually refers to or determines countable nouns in the singular (This box is full. I'll go and get another. Mary's broken her leg so we need another actress for the part.), but like every it may determine countable nouns in the plural meaning a sum of money, a distance, a set of something, etc.: We've got another 20 miles to go. I've got another three questions for you.
The phrase "another three questions" also appears in Donna Williams, Autism: An Inside-Out Approach (1996), Chen Huaailin, "Magic in the Tube: The Impact of Hong Kong Television in Guangzhou," in China Review 1999 (1999), Myriam Callus & Jackie Sykes, English Elements (3) (2004), Roberta Ambrosino, "Faculty Development Programs at State Universities Toward the Goal of Academic Accessibility for Students with Learning Disabilities" (2007), Karen Morrison & Shelley Wicks, FCS Mathematical Literacy, Level 3 (2008), Sergei Abramovich, Topics in Mathematics for Elementary Teachers: A Technology-Enhanced Experiential Approach (2010), Ann Callander & Jacquie Buttriss, Thinking Skills for SEN Learners: Practical Strategies for Developing Thinking and Learning, Arieh Ben-Naim, Discover Entropy and the Second Law of Thermodynamics: A Playful Way of Discovering a Law of Nature (2010), Dianna Booher, Speak with Confidence (2011), Randall Shumaker, Virtual, Augmented and Mixed Reality: Designing and Developing Augmented and Virtual Environments (2013), and James Campbell, Polarized: Making Sense of a Divided America (2016).
In some of these instances, the phrase may have the sense "another set of three questions," as Hjulmand's Contrastive Grammar prescribes. But in others, the phrase simply means "three questions in addition to however many questions are or were already asked." In my opinion, this usage is commonplace in idiomatic English, and treating it as fundamentally incorrect entails denying the validity of an everyday speech pattern among fluent English speakers. That "another three questions" appears fairly often in copyedited publications suggests that no widely respected stricture against such usage prevails in the publishing world.