Though I don't know what third part identity provider you're using, but I think you can achieve add the provider by adding custom providers in custom policies.
First, according to your post , I assume that you're using the Oauth/OIDC provider.
Example: Add LinkedIn as an identity provider by using custom policies:
In the <ClaimsProviders>
element, add the following XML snippet:
<ClaimsProvider>
<Domain>linkedin.com</Domain>
<DisplayName>LinkedIn</DisplayName>
<TechnicalProfiles>
<TechnicalProfile Id="LinkedIn-OAUTH">
<DisplayName>LinkedIn</DisplayName>
<Protocol Name="OAuth2" />
<Metadata>
<Item Key="ProviderName">linkedin</Item>
<Item Key="authorization_endpoint">https://www.linkedin.com/oauth/v2/authorization</Item>
<Item Key="AccessTokenEndpoint">https://www.linkedin.com/oauth/v2/accessToken</Item>
<Item Key="ClaimsEndpoint">https://api.linkedin.com/v1/people/~:(id,first-name,last-name,email-address,headline)</Item>
<Item Key="ClaimsEndpointAccessTokenName">oauth2_access_token</Item>
<Item Key="ClaimsEndpointFormatName">format</Item>
<Item Key="ClaimsEndpointFormat">json</Item>
<Item Key="scope">r_emailaddress r_basicprofile</Item>
<Item Key="HttpBinding">POST</Item>
<Item Key="UsePolicyInRedirectUri">0</Item>
<Item Key="client_id">Your LinkedIn application client ID</Item>
</Metadata>
<CryptographicKeys>
<Key Id="client_secret" StorageReferenceId="B2C_1A_LinkedInSecret" />
</CryptographicKeys>
<OutputClaims>
<OutputClaim ClaimTypeReferenceId="socialIdpUserId" PartnerClaimType="id" />
<OutputClaim ClaimTypeReferenceId="givenName" PartnerClaimType="firstName" />
<OutputClaim ClaimTypeReferenceId="surname" PartnerClaimType="lastName" />
<OutputClaim ClaimTypeReferenceId="email" PartnerClaimType="emailAddress" />
<!--<OutputClaim ClaimTypeReferenceId="jobTitle" PartnerClaimType="headline" />-->
<OutputClaim ClaimTypeReferenceId="identityProvider" DefaultValue="linkedin.com" />
<OutputClaim ClaimTypeReferenceId="authenticationSource" DefaultValue="socialIdpAuthentication" />
</OutputClaims>
<OutputClaimsTransformations>
<OutputClaimsTransformation ReferenceId="CreateRandomUPNUserName" />
<OutputClaimsTransformation ReferenceId="CreateUserPrincipalName" />
<OutputClaimsTransformation ReferenceId="CreateAlternativeSecurityId" />
<OutputClaimsTransformation ReferenceId="CreateSubjectClaimFromAlternativeSecurityId" />
</OutputClaimsTransformations>
<UseTechnicalProfileForSessionManagement ReferenceId="SM-SocialLogin" />
</TechnicalProfile>
</TechnicalProfiles>
</ClaimsProvider>
Also, you can add <Item Key="AccessTokenResponseFormat">json</Item>
to claim json type of endpoint.
You can see more details about Adding LinkedIn as an identity provider by using custom policies in this document.
Additional:
I don't know what third identity provider you're using , if it helps ,please let me know.