diff --git a/goopenssl.c b/goopenssl.c
index 1e428d5..c3385b9 100644
--- a/goopenssl.c
+++ b/goopenssl.c
@@ -35,6 +35,19 @@ FOR_ALL_OPENSSL_FUNCTIONS
 #undef DEFINEFUNC_RENAMED_1_1
 #undef DEFINEFUNC_RENAMED_3_0
 
+// go_openssl_fips_enabled returns 1 if FIPS mode is enabled, 0 otherwise.
+// As a special case, it returns -1 if it cannot determine if FIPS mode is enabled.
+// See openssl.FIPS for details about its implementation.
+//
+// This function is reimplemented here because openssl.FIPS assumes that
+// all the OpenSSL bindings are loaded, that is, go_openssl_load_functions has
+// already been called. On the other hand, go_openssl_fips_enabled is called from
+// openssl.CheckVersion, which is used to check if a given OpenSSL shared library
+// exists and is FIPS compliant. That shared library might not be the one that
+// was passed to go_openssl_load_functions, or it might not even have been called at all.
+//
+// It is written in C because it is not possible to directly call C function pointers
+// retrieved using dlsym from Go.
 int
 go_openssl_fips_enabled(void* handle)
 {
@@ -45,15 +58,24 @@ go_openssl_fips_enabled(void* handle)
         return FIPS_mode();
 
     // For OpenSSL 3.x.
-    int (*EVP_default_properties_is_fips_enabled)(void*);
-    int (*OSSL_PROVIDER_available)(void*, const char*);
-    EVP_default_properties_is_fips_enabled = (int (*)(void*))dlsym(handle, "EVP_default_properties_is_fips_enabled"); 
-    OSSL_PROVIDER_available = (int (*)(void*, const char*))dlsym(handle, "OSSL_PROVIDER_available"); 
-    if (EVP_default_properties_is_fips_enabled != NULL && OSSL_PROVIDER_available != NULL &&
-        EVP_default_properties_is_fips_enabled(NULL) == 1 && OSSL_PROVIDER_available(NULL, "fips") == 1)
-            return 1;
+    int (*EVP_default_properties_is_fips_enabled)(void*) = (int (*)(void*))dlsym(handle, "EVP_default_properties_is_fips_enabled");
+    void *(*EVP_MD_fetch)(void*, const char*, const char*) = (void* (*)(void*, const char*, const char*))dlsym(handle, "EVP_MD_fetch");
+    void (*EVP_MD_free)(void*) = (void (*)(void*))dlsym(handle, "EVP_MD_free");
 
-    return 0;
+    if (EVP_default_properties_is_fips_enabled == NULL || EVP_MD_fetch == NULL || EVP_MD_free == NULL) {
+        // Shouldn't happen, but if it does, we can't determine if FIPS mode is enabled.
+        return -1;
+    }
+
+    if (EVP_default_properties_is_fips_enabled(NULL) != 1)
+        return 0;
+
+    void *md = EVP_MD_fetch(NULL, "SHA2-256", NULL);
+    if (md == NULL)
+        return 0;
+
+    EVP_MD_free(md);
+    return 1;
 }
 
 // Load all the functions stored in FOR_ALL_OPENSSL_FUNCTIONS
diff --git a/openssl.go b/openssl.go
index 1562cee..fb4b65b 100644
--- a/openssl.go
+++ b/openssl.go
@@ -38,8 +38,14 @@ func CheckVersion(version string) (exists, fips bool) {
 		return false, false
 	}
 	defer dlclose(handle)
-	fips = C.go_openssl_fips_enabled(handle) == 1
-	return true, fips
+	enabled := C.go_openssl_fips_enabled(handle)
+	fips = enabled == 1
+	// If go_openssl_fips_enabled returns -1, it means that all or some of the necessary
+	// functions are not available. This can be due to the version of OpenSSL being too old,
+	// too incompatible, or the shared library not being an OpenSSL library. In any case,
+	// we shouldn't consider this library to be valid for our purposes.
+	exists = enabled != -1
+	return
 }
 
 // Init loads and initializes OpenSSL from the shared library at path.
@@ -96,23 +102,36 @@ func VersionText() string {
 var (
 	providerNameFips    = C.CString("fips")
 	providerNameDefault = C.CString("default")
+
+	algorithmSHA256 = C.CString("SHA2-256")
 )
 
-// FIPS returns true if OpenSSL is running in FIPS mode, else returns false.
+// FIPS returns true if OpenSSL is running in FIPS mode and there is
+// a provider available that supports FIPS. It returns false otherwise.
 func FIPS() bool {
 	switch vMajor {
 	case 1:
 		return C.go_openssl_FIPS_mode() == 1
 	case 3:
-		// If FIPS is not enabled via default properties, then we are sure FIPS is not used.
-		if C.go_openssl_EVP_default_properties_is_fips_enabled(nil) == 0 {
+		// Check if the default properties contain `fips=1`.
+		if C.go_openssl_EVP_default_properties_is_fips_enabled(nil) != 1 {
+			// Note that it is still possible that the provider used by default is FIPS-compliant,
+			// but that wouldn't be a system or user requirement.
+			return false
+		}
+		// Check if the SHA-256 algorithm is available. If it is, then we can be sure that there is a provider available that matches
+		// the `fips=1` query. Most notably, this works for the common case of using the built-in FIPS provider.
+		//
+		// Note that this approach has a small chance of false negative if the FIPS provider doesn't provide the SHA-256 algorithm,
+		// but that is highly unlikely because SHA-256 is one of the most common algorithms and fundamental to many cryptographic operations.
+		// It also has a small chance of false positive if the FIPS provider implements the SHA-256 algorithm but not the other algorithms
+		// used by the caller application, but that is also unlikely because the FIPS provider should provide all common algorithms.
+		md := C.go_openssl_EVP_MD_fetch(nil, algorithmSHA256, nil)
+		if md == nil {
 			return false
 		}
-		// EVP_default_properties_is_fips_enabled can return true even if the FIPS provider isn't loaded,
-		// it is only based on the default properties.
-		// We can be sure that the FIPS provider is available if we can fetch an algorithm, e.g., SHA2-256,
-		// explicitly setting `fips=yes`.
-		return C.go_openssl_OSSL_PROVIDER_available(nil, providerNameFips) == 1
+		C.go_openssl_EVP_MD_free(md)
+		return true
 	default:
 		panic(errUnsupportedVersion())
 	}