If a Shimura variety is fully Hodge-Newton decomposable, its mod p reduction enjoys particularly nice properties. These cases are well-understood and characterized by a certain inequality known as the minute condition. In a joint project with Xuhua He and Eva Viehmann, we study the next larger cases, where the minute condition is barely not satisfied any more. I will give a classification of these cases and explain the geometric properties of the corresponding Ekedahl-Oort and Kottwitz-Rapoport strata. The Ekedahl-Oort strata are described by a new class of affine Deligne-Lusztig varieties introduced in a joint work with Ryosuke Shimada and Qingchao Yu. For these affine Deligne-Lusztig varieties of positive Coxter type, we give a full description of their dimensions and the sets of irreducible components.