Complex asymmetric 3D structures are observed in the outflows of evolved low- and intermediate-mass stars, and are believed to be shaped through the interaction of companions that remain hidden within the dense wind. One example is the AGB star R Aql, for which ALMA (Atacama Large Millimeter Array) observations have revealed complex wind structures that might originate from a higher-order stellar system. We investigate how triple systems can shape the outflow of asymptotic giant branch (AGB) stars and characterise the different wind structures that form. For simplicity, we solely focus on co-planar systems in a hierarchical, stable orbit, consisting of an AGB star with one relatively close companion, and another one at a large orbital separation. We modelled a grid of hierarchical triple systems including a wind-launching AGB star, with the smoothed-particle-hydrodynamic Phantom code. We varied the outer companion mass, the AGB wind velocity, and the orbital eccentricities to study the impact of these parameters on the wind morphology. To study the impact of adding a triple companion, we additionally modelled and analysed a small grid of binary sub-systems, for comparison. To investigate if R Aql could be shaped by a triple system, we post-processed one of our triple models with a radiative transfer routine, and compared this to data of the ALMA ATOMIUM programme. The characteristic wind structures resulting from a hierarchical triple system are the following. A large two-edged spiral wake results behind the outer companion star. This structure lies on top of the spiral structure formed by the close binary, which is itself affected by the orbital motion around the system's centre of mass, such that it resembles a snail-shell pattern. This dense inner spiral pattern interacts with, and strongly impacts, the spiral wake of the outer companion, resulting in a wave pattern on the outer edge of this spiral wake. The higher the mass of the outer companion, the larger the density enhancement and the more radially compressed the outer spiral. Lowering the wind velocity has a similar effect, and additionally results in an elongation of the global wind morphology. Introducing eccentricity in the inner and outer orbit of the hierarchical system results in complex phase-dependent wind-companion interactions, and consequently in asymmetries in the inner part of the wind and the global morphology, respectively. From the comparison of our models to the observations of R Aql, we conclude that this circumstellar environment might be shaped by a similar system to the ones modelled in this work, but an elaborate study of the observational data is needed to better determine the orbital parameters and characteristics of the central system. The modelled outflow of an AGB star in a co-planar hierarchical systems is characterised by a large-scale spiral wake with a wavey outer edge, attached to the outer companion, on top of a compact inner spiral pattern that resembles a snail-shell pattern.