Try adding layout support to Cover block.

lib/block-supports/layout.php

@@ -284,6 +284,29 @@ function gutenberg_get_layout_style( $selector, $layout, $has_block_gap_support
 	return '';
 }
 
+/**
+ * Adds an identifying classname to the inner block wrapper.
+ *
+ * @param array $parsed_block A parsed block object.
+ * @return array              The updated block object.
+ */
+function gutenberg_identify_inner_block_wrapper( $parsed_block ) {
+
+	if ( ! isset( $parsed_block['innerContent'][0] ) ) {
+		return $parsed_block;
+	}
+
+	$inner_class_position = strrpos( $parsed_block['innerContent'][0], 'class="' );
+
+	if ( ! $inner_class_position ) {
+		return $parsed_block;
+	}
+
+	$parsed_block['innerContent'][0] = substr_replace( $parsed_block['innerContent'][0], 'class="is-inner-block-wrapper ', $inner_class_position, strlen( 'class="' ) );
+
+	return $parsed_block;
+}
+
 /**
  * Renders the layout config to the block wrapper.
  *
@@ -387,14 +410,25 @@ function gutenberg_render_layout_support_flag( $block_content, $block ) {
 		}
 	}
 
-	// This assumes the hook only applies to blocks with a single wrapper.
-	// I think this is a reasonable limitation for that particular hook.
-	$content = preg_replace(
-		'/' . preg_quote( 'class="', '/' ) . '/',
-		'class="' . esc_attr( implode( ' ', $class_names ) ) . ' ',
-		$block_content,
-		1
-	);
+	// Ideally we'll be able to attach this class to all inner block wrappers and
+	// we won't need the condition or the fallback.
+	if ( strpos( $block_content, 'is-inner-block-wrapper' ) ) {
+		$content = preg_replace(
+			'/' . preg_quote( 'is-inner-block-wrapper', '/' ) . '/',
+			esc_attr( implode( ' ', $class_names ) ),
+			$block_content,
+			1
+		);
+	} else {
+		// This assumes the hook only applies to blocks with a single wrapper.
+		// I think this is a reasonable limitation for that particular hook.
+		$content = preg_replace(
+			'/' . preg_quote( 'class="', '/' ) . '/',
+			'class="' . esc_attr( implode( ' ', $class_names ) ) . ' ',
+			$block_content,
+			1
+		);
+	}
 
 	return $content;
 }
@@ -406,6 +440,12 @@ function gutenberg_render_layout_support_flag( $block_content, $block ) {
 		'register_attribute' => 'gutenberg_register_layout_support',
 	)
 );
+
+if ( function_exists( 'wp_identify_inner_block_wrapper' ) ) {
+	remove_filter( 'render_block', 'wp_identify_inner_block_wrapper' );
+}
+add_filter( 'render_block_data', 'gutenberg_identify_inner_block_wrapper' );
+
 if ( function_exists( 'wp_render_layout_support_flag' ) ) {
 	remove_filter( 'render_block', 'wp_render_layout_support_flag' );
 }

packages/block-library/src/cover/block.json

@@ -105,6 +105,13 @@
 			"__experimentalDefaultControls": {
 				"fontSize": true
 			}
+		},
+		"__experimentalLayout": {
+			"allowSwitching": true,
+			"allowInheriting": false,
+			"default": {
+				"type": "flow"
+			}
 		}
 	},
 	"editorStyle": "wp-block-cover-editor",