diff --git a/t/t0061-run-command.sh b/t/t0061-run-command.sh
index 7b5423eebd..e2411f6a9b 100755
--- a/t/t0061-run-command.sh
+++ b/t/t0061-run-command.sh
@@ -130,7 +130,8 @@ World
 EOF
 
 test_expect_success 'run_command runs in parallel with more jobs available than tasks' '
-	test-tool run-command run-command-parallel 5 sh -c "printf \"%s\n%s\n\" Hello World" 2>actual &&
+	test-tool run-command run-command-parallel 5 sh -c "printf \"%s\n%s\n\" Hello World" >out 2>actual &&
+	test_must_be_empty out &&
 	test_cmp expect actual
 '
 
@@ -141,7 +142,8 @@ test_expect_success 'run_command runs ungrouped in parallel with more jobs avail
 '
 
 test_expect_success 'run_command runs in parallel with as many jobs as tasks' '
-	test-tool run-command run-command-parallel 4 sh -c "printf \"%s\n%s\n\" Hello World" 2>actual &&
+	test-tool run-command run-command-parallel 4 sh -c "printf \"%s\n%s\n\" Hello World" >out 2>actual &&
+	test_must_be_empty out &&
 	test_cmp expect actual
 '
 
@@ -152,7 +154,8 @@ test_expect_success 'run_command runs ungrouped in parallel with as many jobs as
 '
 
 test_expect_success 'run_command runs in parallel with more tasks than jobs available' '
-	test-tool run-command run-command-parallel 3 sh -c "printf \"%s\n%s\n\" Hello World" 2>actual &&
+	test-tool run-command run-command-parallel 3 sh -c "printf \"%s\n%s\n\" Hello World" >out 2>actual &&
+	test_must_be_empty out &&
 	test_cmp expect actual
 '
 
@@ -172,7 +175,8 @@ asking for a quick stop
 EOF
 
 test_expect_success 'run_command is asked to abort gracefully' '
-	test-tool run-command run-command-abort 3 false 2>actual &&
+	test-tool run-command run-command-abort 3 false >out 2>actual &&
+	test_must_be_empty out &&
 	test_cmp expect actual
 '
 
@@ -187,7 +191,8 @@ no further jobs available
 EOF
 
 test_expect_success 'run_command outputs ' '
-	test-tool run-command run-command-no-jobs 3 sh -c "printf \"%s\n%s\n\" Hello World" 2>actual &&
+	test-tool run-command run-command-no-jobs 3 sh -c "printf \"%s\n%s\n\" Hello World" >out 2>actual &&
+	test_must_be_empty out &&
 	test_cmp expect actual
 '
 
diff --git a/t/t1800-hook.sh b/t/t1800-hook.sh
index 43fcb7c0bf..2ef3579fa7 100755
--- a/t/t1800-hook.sh
+++ b/t/t1800-hook.sh
@@ -95,7 +95,7 @@ test_expect_success 'git hook run -- out-of-repo runs excluded' '
 test_expect_success 'git -c core.hooksPath=<PATH> hook run' '
 	mkdir my-hooks &&
 	write_script my-hooks/test-hook <<-\EOF &&
-	echo Hook ran $1 >>actual
+	echo Hook ran $1
 	EOF
 
 	cat >expect <<-\EOF &&
diff --git a/t/t5526-fetch-submodules.sh b/t/t5526-fetch-submodules.sh
index 75da8acf8f..b9546ef8e5 100755
--- a/t/t5526-fetch-submodules.sh
+++ b/t/t5526-fetch-submodules.sh
@@ -178,6 +178,7 @@ test_expect_success "submodule.recurse option triggers recursive fetch" '
 '
 
 test_expect_success "fetch --recurse-submodules -j2 has the same output behaviour" '
+	test_when_finished "rm -f trace.out" &&
 	add_submodule_commits &&
 	(
 		cd downstream &&
@@ -705,15 +706,22 @@ test_expect_success "'fetch.recurseSubmodules=on-demand' works also without .git
 
 test_expect_success 'fetching submodules respects parallel settings' '
 	git config fetch.recurseSubmodules true &&
+	test_when_finished "rm -f downstream/trace.out" &&
 	(
 		cd downstream &&
 		GIT_TRACE=$(pwd)/trace.out git fetch &&
 		grep "1 tasks" trace.out &&
+		>trace.out &&
+
 		GIT_TRACE=$(pwd)/trace.out git fetch --jobs 7 &&
 		grep "7 tasks" trace.out &&
+		>trace.out &&
+
 		git config submodule.fetchJobs 8 &&
 		GIT_TRACE=$(pwd)/trace.out git fetch &&
 		grep "8 tasks" trace.out &&
+		>trace.out &&
+
 		GIT_TRACE=$(pwd)/trace.out git fetch --jobs 9 &&
 		grep "9 tasks" trace.out &&
 		>trace.out &&