diff --git a/internal/lsp/source/completion.go b/internal/lsp/source/completion.go
index f1f50dd936..a2242aa552 100644
--- a/internal/lsp/source/completion.go
+++ b/internal/lsp/source/completion.go
@@ -1375,8 +1375,6 @@ Nodes:
 						} else {
 							inf.objType = sig.Params().At(exprIdx).Type()
 						}
-
-						break Nodes
 					}
 				}
 
@@ -1386,14 +1384,10 @@ Nodes:
 					if obj != nil && obj.Parent() == types.Universe {
 						inf.objKind |= c.builtinArgKind(obj, node)
 
-						if obj.Name() == "new" {
-							inf.typeName.wantTypeName = true
-						}
-
 						// Defer call to builtinArgType so we can provide it the
 						// inferred type from its parent node.
 						defer func() {
-							inf.objType, inf.variadic = c.builtinArgType(obj, node, inf.objType)
+							inf.objType, inf.typeName.wantTypeName, inf.variadic = c.builtinArgType(obj, node, inf.objType)
 						}()
 
 						// The expected type of builtin arguments like append() is
diff --git a/internal/lsp/source/completion_builtin.go b/internal/lsp/source/completion_builtin.go
index da2c5f8ff8..f6046c55be 100644
--- a/internal/lsp/source/completion_builtin.go
+++ b/internal/lsp/source/completion_builtin.go
@@ -51,7 +51,7 @@ func (c *completer) builtinArgKind(obj types.Object, call *ast.CallExpr) objKind
 // builtinArgType infers the type of an argument to a builtin
 // function. "parentType" is the inferred type for the builtin call's
 // parent node.
-func (c *completer) builtinArgType(obj types.Object, call *ast.CallExpr, parentType types.Type) (infType types.Type, variadic bool) {
+func (c *completer) builtinArgType(obj types.Object, call *ast.CallExpr, parentType types.Type) (infType types.Type, wantType, variadic bool) {
 	exprIdx := exprAtPos(c.pos, call.Args)
 
 	switch obj.Name() {
@@ -93,6 +93,7 @@ func (c *completer) builtinArgType(obj types.Object, call *ast.CallExpr, parentT
 			infType = t2
 		}
 	case "new":
+		wantType = true
 		if parentType != nil {
 			// Expected type for "new" is the de-pointered parent type.
 			if ptr, ok := parentType.Underlying().(*types.Pointer); ok {
@@ -101,9 +102,12 @@ func (c *completer) builtinArgType(obj types.Object, call *ast.CallExpr, parentT
 		}
 	case "make":
 		if exprIdx == 0 {
+			wantType = true
 			infType = parentType
+		} else {
+			infType = types.Typ[types.Int]
 		}
 	}
 
-	return infType, variadic
+	return infType, wantType, variadic
 }
diff --git a/internal/lsp/testdata/builtins/builtin_args.go b/internal/lsp/testdata/builtins/builtin_args.go
index 040d7e396a..8b3d10a46a 100644
--- a/internal/lsp/testdata/builtins/builtin_args.go
+++ b/internal/lsp/testdata/builtins/builtin_args.go
@@ -12,6 +12,12 @@ func _() {
 		aInt      int            //@item(builtinInt, "aInt", "int", "var")
 	)
 
+	type (
+		aSliceType []int          //@item(builtinSliceType, "aSliceType", "[]int", "type")
+		aChanType  chan int       //@item(builtinChanType, "aChanType", "chan int", "type")
+		aMapType   map[string]int //@item(builtinMapType, "aMapType", "map[string]int", "type")
+	)
+
 	close() //@rank(")", builtinChan, builtinSlice)
 
 	append() //@rank(")", builtinSlice, builtinChan)
@@ -27,9 +33,10 @@ func _() {
 
 	cap() //@rank(")", builtinSlice, builtinMap),rank(")", builtinArray, builtinString),rank(")", builtinArrayPtr, builtinPtr),rank(")", builtinChan, builtinInt)
 
-	make() //@rank(")", builtinMap, builtinInt),rank(")", builtinChan, builtinInt),rank(")", builtinSlice, builtinInt)
+	make()              //@rank(")", builtinMapType, int),rank(")", builtinChanType, int),rank(")", builtinSliceType, int),rank(")", builtinMapType, builtinMap)
+	make(aSliceType, a) //@rank(")", builtinInt, builtinSlice)
 
-	var _ []int = make() //@rank(")", builtinSlice, builtinMap)
+	var _ []int = make() //@rank(")", builtinSliceType, builtinSlice)
 
 	type myStruct struct{}  //@item(builtinStructType, "myStruct", "struct{...}", "struct")
 	new()                   //@rank(")", builtinStructType, builtinInt)
diff --git a/internal/lsp/testdata/summary.txt.golden b/internal/lsp/testdata/summary.txt.golden
index bbb39094cd..e85b9f839b 100644
--- a/internal/lsp/testdata/summary.txt.golden
+++ b/internal/lsp/testdata/summary.txt.golden
@@ -4,7 +4,7 @@ CompletionSnippetCount = 66
 UnimportedCompletionsCount = 9
 DeepCompletionsCount = 5
 FuzzyCompletionsCount = 8
-RankedCompletionsCount = 66
+RankedCompletionsCount = 68
 CaseSensitiveCompletionsCount = 4
 DiagnosticsCount = 37
 FoldingRangesCount = 2