diff --git a/src/core/evaluator.js b/src/core/evaluator.js
index e1badcec3670b..4f33989d0757e 100644
--- a/src/core/evaluator.js
+++ b/src/core/evaluator.js
@@ -4664,9 +4664,22 @@ class PartialEvaluator {
       throw new FormatError("invalid font name");
     }
 
-    let fontFile, subtype, length1, length2, length3;
+    let fontFile, fontFileN, subtype, length1, length2, length3;
     try {
-      fontFile = descriptor.get("FontFile", "FontFile2", "FontFile3");
+      fontFile = descriptor.get("FontFile");
+      if (fontFile) {
+        fontFileN = 1;
+      } else {
+        fontFile = descriptor.get("FontFile2");
+        if (fontFile) {
+          fontFileN = 2;
+        } else {
+          fontFile = descriptor.get("FontFile3");
+          if (fontFile) {
+            fontFileN = 3;
+          }
+        }
+      }
 
       if (fontFile) {
         if (!(fontFile instanceof BaseStream)) {
@@ -4776,6 +4789,7 @@ class PartialEvaluator {
       name: fontName.name,
       subtype,
       file: fontFile,
+      fontFileN,
       length1,
       length2,
       length3,
diff --git a/src/core/fonts.js b/src/core/fonts.js
index 71c5f3eac60e3..8cf66e1311693 100644
--- a/src/core/fonts.js
+++ b/src/core/fonts.js
@@ -2682,10 +2682,23 @@ class Font {
     }
 
     const isTrueType = !tables["CFF "];
+    let parsedCff = null;
     if (!isTrueType) {
+      try {
+        parsedCff = new CFFParser(
+          new Stream(tables["CFF "].data),
+          properties,
+          SEAC_ANALYSIS_ENABLED
+        ).parse();
+      } catch {
+        warn("Failed to parse font " + properties.loadedName);
+      }
+
       // OpenType font (skip composite fonts with non-default glyph mapping).
       if (
-        (header.version === "OTTO" && !properties.composite) ||
+        (header.version === "OTTO" &&
+          (!properties.composite ||
+            (properties.fontFileN === 3 && parsedCff?.isCIDFont))) ||
         !tables.head ||
         !tables.hhea ||
         !tables.maxp ||
@@ -2725,19 +2738,11 @@ class Font {
     }
 
     let numGlyphsFromCFF;
-    if (!isTrueType) {
+    if (parsedCff) {
       try {
-        // Trying to repair CFF file
-        const parser = new CFFParser(
-          new Stream(tables["CFF "].data),
-          properties,
-          SEAC_ANALYSIS_ENABLED
-        );
-        const cff = parser.parse();
-        cff.duplicateFirstGlyph();
-        const compiler = new CFFCompiler(cff);
-        tables["CFF "].data = compiler.compile();
-        numGlyphsFromCFF = cff.charStringCount;
+        parsedCff.duplicateFirstGlyph();
+        tables["CFF "].data = new CFFCompiler(parsedCff).compile();
+        numGlyphsFromCFF = parsedCff.charStringCount;
       } catch {
         warn("Failed to compile font " + properties.loadedName);
       }
diff --git a/test/pdfs/.gitignore b/test/pdfs/.gitignore
index a0c521f326746..bc6027dc6a5b4 100644
--- a/test/pdfs/.gitignore
+++ b/test/pdfs/.gitignore
@@ -921,3 +921,4 @@
 !operator_list_cycle.pdf
 !knockout_groups_test.pdf
 !issue18032.pdf
+!Embedded_font.pdf
diff --git a/test/pdfs/Embedded_font.pdf b/test/pdfs/Embedded_font.pdf
new file mode 100644
index 0000000000000..823909f92a813
Binary files /dev/null and b/test/pdfs/Embedded_font.pdf differ
diff --git a/test/test_manifest.json b/test/test_manifest.json
index adc5211c63109..e3d03daa37e93 100644
--- a/test/test_manifest.json
+++ b/test/test_manifest.json
@@ -14260,5 +14260,12 @@
     "rounds": 1,
     "type": "eq",
     "about": "Non-isolated knockout group with a nested non-isolated subgroup over existing text."
+  },
+  {
+    "id": "Embedded_font.pdf",
+    "file": "pdfs/Embedded_font.pdf",
+    "md5": "b68dd5a3e6833d1af94e295fe1d60285",
+    "rounds": 1,
+    "type": "eq"
   }
 ]