diff --git a/Makefile b/Makefile
index 7a5748f5ae..8f16aa9a02 100644
--- a/Makefile
+++ b/Makefile
@@ -1,30 +1,8 @@
 .PHONY: all binary build cross default docs docs-build docs-shell shell test test-docker-py test-integration-cli test-unit validate
 
 # get OS/Arch of docker engine
-DOCKER_OSARCH := $(shell bash -c 'source hack/make/.detect-daemon-osarch && echo $${DOCKER_ENGINE_OSARCH:+$$DOCKER_CLIENT_OSARCH}')
-# default for linux/amd64 and others
-DOCKERFILE := Dockerfile
-# switch to different Dockerfile for linux/arm
-ifeq ($(DOCKER_OSARCH), linux/arm)
-	DOCKERFILE := Dockerfile.armhf
-else
-ifeq ($(DOCKER_OSARCH), linux/arm64)
-	DOCKERFILE := Dockerfile.aarch64
-else
-ifeq ($(DOCKER_OSARCH), linux/ppc64le)
-	DOCKERFILE := Dockerfile.ppc64le
-else
-ifeq ($(DOCKER_OSARCH), linux/s390x)
-	DOCKERFILE := Dockerfile.s390x
-else
-ifeq ($(DOCKER_OSARCH), windows/amd64)
-	DOCKERFILE := Dockerfile.windows
-endif
-endif
-endif
-endif
-endif
-export DOCKERFILE
+DOCKER_OSARCH := $(shell bash -c 'source hack/make/.detect-daemon-osarch && echo $${DOCKER_ENGINE_OSARCH:-$$DOCKER_CLIENT_OSARCH}')
+DOCKERFILE := $(shell bash -c 'source hack/make/.detect-daemon-osarch && echo $${DOCKERFILE}')
 
 # env vars passed through directly to Docker's build scripts
 # to allow things like `make DOCKER_CLIENTONLY=1 binary` easily
@@ -37,7 +15,6 @@ DOCKER_ENVS := \
 	-e DOCKER_CLIENTONLY \
 	-e DOCKER_DEBUG \
 	-e DOCKER_EXPERIMENTAL \
-	-e DOCKERFILE \
 	-e DOCKER_GRAPHDRIVER \
 	-e DOCKER_INCREMENTAL_BINARY \
 	-e DOCKER_REMAP_ROOT \
diff --git a/hack/make/.detect-daemon-osarch b/hack/make/.detect-daemon-osarch
index 5c6d3f5783..571e802eb9 100644
--- a/hack/make/.detect-daemon-osarch
+++ b/hack/make/.detect-daemon-osarch
@@ -1,34 +1,66 @@
 #!/bin/bash
 set -e
 
+docker-version-osarch() {
+	local target="$1" # "Client" or "Server"
+	local fmtStr="{{.${target}.Os}}/{{.${target}.Arch}}"
+	if docker version -f "$fmtStr" 2>/dev/null; then
+		# if "docker version -f" works, let's just use that!
+		return
+	fi
+	docker version | awk '
+		$1 ~ /^(Client|Server):$/ { section = 0 }
+		$1 == "'"$target"':" { section = 1; next }
+		section && $1 == "OS/Arch:" { print $2 }
+
+		# old versions of Docker
+		$1 == "OS/Arch" && $2 == "('"${target,,}"'):" { print $3 }
+	'
+}
+
 # Retrieve OS/ARCH of docker daemon, eg. linux/amd64
-export DOCKER_ENGINE_OSARCH="$(docker version | awk '
-	$1 == "Client:" { server = 0; next }
-	$1 == "Server:" { server = 1; next }
-	server && $1 == "OS/Arch:" { print $2 }
-')"
+export DOCKER_ENGINE_OSARCH="$(docker-version-osarch 'Server')"
 export DOCKER_ENGINE_GOOS="${DOCKER_ENGINE_OSARCH%/*}"
 export DOCKER_ENGINE_GOARCH="${DOCKER_ENGINE_OSARCH##*/}"
 DOCKER_ENGINE_GOARCH=${DOCKER_ENGINE_GOARCH:=amd64}
 
 # and the client, just in case
-export DOCKER_CLIENT_OSARCH="$(docker version | awk '
-	$1 == "Client:" { client = 1; next }
-	$1 == "Server:" { client = 0; next }
-	client && $1 == "OS/Arch:" { print $2 }
-')"
+export DOCKER_CLIENT_OSARCH="$(docker-version-osarch 'Client')"
+export DOCKER_CLIENT_GOOS="${DOCKER_CLIENT_OSARCH%/*}"
+export DOCKER_CLIENT_GOARCH="${DOCKER_CLIENT_OSARCH##*/}"
+DOCKER_CLIENT_GOARCH=${DOCKER_CLIENT_GOARCH:=amd64}
 
 # Retrieve the architecture used in contrib/builder/(deb|rpm)/$PACKAGE_ARCH/
-PACKAGE_ARCH="amd64"
-case "$DOCKER_ENGINE_OSARCH" in
-	linux/arm)
+PACKAGE_ARCH='amd64'
+case "${DOCKER_ENGINE_GOARCH:-$DOCKER_CLIENT_GOARCH}" in
+	arm)
 		PACKAGE_ARCH='armhf'
 		;;
-	linux/ppc64le)
-		PACKAGE_ARCH='ppc64le'
+	arm64)
+		PACKAGE_ARCH='aarch64'
 		;;
-	linux/s390x)
-		PACKAGE_ARCH='s390x'
+	amd64|ppc64le|s390x)
+		PACKAGE_ARCH="${DOCKER_ENGINE_GOARCH:-$DOCKER_CLIENT_GOARCH}"
+		;;
+	*)
+		echo >&2 "warning: not sure how to convert '$DOCKER_ENGINE_GOARCH' to a 'Docker' arch, assuming '$PACKAGE_ARCH'"
 		;;
 esac
 export PACKAGE_ARCH
+
+DOCKERFILE='Dockerfile'
+TEST_IMAGE_NAMESPACE=
+case "$PACKAGE_ARCH" in
+	amd64)
+		case "${DOCKER_ENGINE_GOOS:-$DOCKER_CLIENT_GOOS}" in
+			windows)
+				DOCKERFILE='Dockerfile.windows'
+				;;
+		esac
+		;;
+	*)
+		DOCKERFILE="Dockerfile.$PACKAGE_ARCH"
+		TEST_IMAGE_NAMESPACE="$PACKAGE_ARCH"
+		;;
+esac
+export DOCKERFILE TEST_IMAGE_NAMESPACE
diff --git a/hack/make/.ensure-frozen-images b/hack/make/.ensure-frozen-images
index 9eaa7f3a4d..d0c55e50cc 100644
--- a/hack/make/.ensure-frozen-images
+++ b/hack/make/.ensure-frozen-images
@@ -9,25 +9,9 @@ images=(
 	hello-world:latest
 )
 
-imagePrefix=
-case "$DOCKER_ENGINE_OSARCH" in
-	linux/arm)
-		imagePrefix='armhf'
-		;;
-	linux/arm64)
-		imagePrefix='aarch64'
-		;;
-	linux/ppc64le)
-		imagePrefix='ppc64le'
-		;;
-	linux/s390x)
-		imagePrefix='s390x'
-		;;
-esac
-
-if [ "$imagePrefix" ]; then
+if [ "$TEST_IMAGE_NAMESPACE" ]; then
 	for (( i = 0; i < ${#images[@]}; i++ )); do
-		images[$i]="$imagePrefix/${images[$i]}"
+		images[$i]="$TEST_IMAGE_NAMESPACE/${images[$i]}"
 	done
 fi
 
@@ -58,7 +42,7 @@ if ! docker inspect "${images[@]}" &> /dev/null; then
 					inCont = 0;
 				}
 			}
-		' "${DOCKERFILE:=Dockerfile}" | sh -x
+		' "$DOCKERFILE" | sh -x
 		# Do not use a subshell for the following command. Windows to Linux CI
 		# runs bash 3.x so will not trap an error in a subshell.
 		# http://stackoverflow.com/questions/22630363/how-does-set-e-work-with-subshells
@@ -66,9 +50,9 @@ if ! docker inspect "${images[@]}" &> /dev/null; then
 	fi
 fi
 
-if [ "$imagePrefix" ]; then
+if [ "$TEST_IMAGE_NAMESPACE" ]; then
 	for image in "${images[@]}"; do
-		target="${image#$imagePrefix/}"
+		target="${image#$TEST_IMAGE_NAMESPACE/}"
 		if [ "$target" != "$image" ]; then
 			# tag images to ensure that all integrations work with the defined image names
 			docker tag "$image" "$target"